| L(s) = 1 | − 2-s − 2·3-s − 3·4-s − 5-s + 2·6-s − 7-s + 5·8-s + 3·9-s + 10-s + 6·12-s + 4·13-s + 14-s + 2·15-s − 10·17-s − 3·18-s + 9·19-s + 3·20-s + 2·21-s − 2·23-s − 10·24-s + 5·25-s − 4·26-s − 10·27-s + 3·28-s + 29-s − 2·30-s + 7·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 3/2·4-s − 0.447·5-s + 0.816·6-s − 0.377·7-s + 1.76·8-s + 9-s + 0.316·10-s + 1.73·12-s + 1.10·13-s + 0.267·14-s + 0.516·15-s − 2.42·17-s − 0.707·18-s + 2.06·19-s + 0.670·20-s + 0.436·21-s − 0.417·23-s − 2.04·24-s + 25-s − 0.784·26-s − 1.92·27-s + 0.566·28-s + 0.185·29-s − 0.365·30-s + 1.25·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5128590124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5128590124\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_4\times C_2$ | \( 1 + T + p^{2} T^{2} + p T^{3} + 9 T^{4} + p^{2} T^{5} + p^{4} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 3 | $C_2^2:C_4$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + 19 T^{4} + 2 p^{2} T^{5} + p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_4\times C_2$ | \( 1 + T - 4 T^{2} - 9 T^{3} + 11 T^{4} - 9 p T^{5} - 4 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 4 T + 3 T^{2} - 50 T^{3} + 341 T^{4} - 50 p T^{5} + 3 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 + 10 T + 83 T^{2} + 460 T^{3} + 2189 T^{4} + 460 p T^{5} + 83 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 9 T + 12 T^{2} + 163 T^{3} - 1095 T^{4} + 163 p T^{5} + 12 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + T + 15 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2:C_4$ | \( 1 - T - 13 T^{2} + 137 T^{3} + 440 T^{4} + 137 p T^{5} - 13 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 7 T - 12 T^{2} + 121 T^{3} + 125 T^{4} + 121 p T^{5} - 12 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 - 8 T + 27 T^{2} - 340 T^{3} + 3401 T^{4} - 340 p T^{5} + 27 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_4\times C_2$ | \( 1 + 25 T + 334 T^{2} + 3125 T^{3} + 22431 T^{4} + 3125 p T^{5} + 334 p^{2} T^{6} + 25 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 5 T - 9 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 3 T - 13 T^{2} + 285 T^{3} + 136 T^{4} + 285 p T^{5} - 13 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 - T - 37 T^{2} + 305 T^{3} + 1976 T^{4} + 305 p T^{5} - 37 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 7 T + 65 T^{2} - 667 T^{3} + 8084 T^{4} - 667 p T^{5} + 65 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 9 T - 15 T^{2} + 629 T^{3} - 4236 T^{4} + 629 p T^{5} - 15 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + T + 73 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 23 T + 133 T^{2} + 1649 T^{3} - 28620 T^{4} + 1649 p T^{5} + 133 p^{2} T^{6} - 23 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 - 7 T + 86 T^{2} - 611 T^{3} + 2239 T^{4} - 611 p T^{5} + 86 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 + 5 T - 19 T^{2} - 635 T^{3} - 1004 T^{4} - 635 p T^{5} - 19 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 11 T + 68 T^{2} + 1215 T^{3} + 17441 T^{4} + 1215 p T^{5} + 68 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 7 T + 179 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_4\times C_2$ | \( 1 + 7 T - 48 T^{2} - 1015 T^{3} - 2449 T^{4} - 1015 p T^{5} - 48 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.13453814989200619846091125470, −6.96688000346556960398001353616, −6.85213373157628456335935708550, −6.78224098764811026254613380669, −6.55013668552136941938210371744, −6.11898301686652276964176812411, −6.02103681736437245400586482651, −5.55053020739334548784779108391, −5.37445183059093223170753909743, −5.23172099993617298459764599431, −4.97175398881160938886336324459, −4.62253285504884315007450757533, −4.46748521241026100364736582094, −4.38971530763066456542235999331, −4.09126376366928408932363361150, −3.55296011489759343441090880759, −3.49899806766795346378556182365, −3.34568543665521335197328324018, −2.86963044473305360334024846000, −2.16299122025374948288107919916, −2.12344364806141788633931316305, −1.60098847918149235423251794340, −1.14789614601125807532343962300, −0.53107794418823455890255233719, −0.48496774589875114370107017072,
0.48496774589875114370107017072, 0.53107794418823455890255233719, 1.14789614601125807532343962300, 1.60098847918149235423251794340, 2.12344364806141788633931316305, 2.16299122025374948288107919916, 2.86963044473305360334024846000, 3.34568543665521335197328324018, 3.49899806766795346378556182365, 3.55296011489759343441090880759, 4.09126376366928408932363361150, 4.38971530763066456542235999331, 4.46748521241026100364736582094, 4.62253285504884315007450757533, 4.97175398881160938886336324459, 5.23172099993617298459764599431, 5.37445183059093223170753909743, 5.55053020739334548784779108391, 6.02103681736437245400586482651, 6.11898301686652276964176812411, 6.55013668552136941938210371744, 6.78224098764811026254613380669, 6.85213373157628456335935708550, 6.96688000346556960398001353616, 7.13453814989200619846091125470
