| L(s) = 1 | − 3·2-s + 2·3-s + 3·4-s + 3·5-s − 6·6-s + 9-s − 9·10-s + 6·12-s + 14·13-s + 6·15-s − 2·16-s − 6·17-s − 3·18-s + 9·20-s − 6·23-s − 3·25-s − 42·26-s + 4·27-s + 9·29-s − 18·30-s + 30·31-s − 6·32-s + 18·34-s + 3·36-s − 24·37-s + 28·39-s + 18·41-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.15·3-s + 3/2·4-s + 1.34·5-s − 2.44·6-s + 1/3·9-s − 2.84·10-s + 1.73·12-s + 3.88·13-s + 1.54·15-s − 1/2·16-s − 1.45·17-s − 0.707·18-s + 2.01·20-s − 1.25·23-s − 3/5·25-s − 8.23·26-s + 0.769·27-s + 1.67·29-s − 3.28·30-s + 5.38·31-s − 1.06·32-s + 3.08·34-s + 1/2·36-s − 3.94·37-s + 4.48·39-s + 2.81·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.064708735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.064708735\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + p T^{2} )^{2}( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} ) \) |
| 3 | $D_{4}$ | \( ( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 5 | $D_4\times C_2$ | \( 1 - 3 T + 12 T^{2} - 27 T^{3} + 71 T^{4} - 27 p T^{5} + 12 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 27 T^{2} + 377 T^{4} - 27 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 31 T^{2} + 537 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T + 2 T^{2} - 72 T^{3} - 201 T^{4} - 72 p T^{5} + 2 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 9 T + 8 T^{2} - 135 T^{3} + 2139 T^{4} - 135 p T^{5} + 8 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 18 T + 210 T^{2} - 1836 T^{3} + 13151 T^{4} - 1836 p T^{5} + 210 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 5 T - 20 T^{2} - 205 T^{3} - 899 T^{4} - 205 p T^{5} - 20 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 24 T + 327 T^{2} - 3240 T^{3} + 25040 T^{4} - 3240 p T^{5} + 327 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 6 T + 5 T^{2} + 450 T^{3} - 3756 T^{4} + 450 p T^{5} + 5 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 6 T + 21 T^{2} + 54 T^{3} - 2692 T^{4} + 54 p T^{5} + 21 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 2 T - 66 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 71 | $D_4\times C_2$ | \( 1 - 6 T + 150 T^{2} - 828 T^{3} + 14855 T^{4} - 828 p T^{5} + 150 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + 6 T + 85 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 270 T^{2} + 31667 T^{4} - 270 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 33 T + 588 T^{2} - 7425 T^{3} + 75011 T^{4} - 7425 p T^{5} + 588 p^{2} T^{6} - 33 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 39 T + 812 T^{2} + 11895 T^{3} + 132795 T^{4} + 11895 p T^{5} + 812 p^{2} T^{6} + 39 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.059837860544257229439088438684, −7.65052726775847318673802253791, −7.01797988008727037178015339679, −6.97267993423042363951242290168, −6.90956719701582355528530522475, −6.31290557489062687614342156901, −6.31027458480504666287269709440, −6.01571060741293661110360917894, −5.84295557175905360746108664247, −5.83910227222597428154607054059, −5.33606203996878601312637995115, −4.77251483431949154101224655195, −4.57006767846061637399974969783, −4.44343398440299428151454173163, −3.99539365037331035321988627286, −3.72216751480771481919384553930, −3.59456460822967411823273910689, −3.01559399350518113244723723505, −2.89304293312412584816532785551, −2.42799224242947469819032395164, −1.99830548317027371002894820121, −1.98924295628334234159921066381, −1.28545071516219709894410587161, −0.941230193954480565229120074302, −0.74887727119996927461514584847,
0.74887727119996927461514584847, 0.941230193954480565229120074302, 1.28545071516219709894410587161, 1.98924295628334234159921066381, 1.99830548317027371002894820121, 2.42799224242947469819032395164, 2.89304293312412584816532785551, 3.01559399350518113244723723505, 3.59456460822967411823273910689, 3.72216751480771481919384553930, 3.99539365037331035321988627286, 4.44343398440299428151454173163, 4.57006767846061637399974969783, 4.77251483431949154101224655195, 5.33606203996878601312637995115, 5.83910227222597428154607054059, 5.84295557175905360746108664247, 6.01571060741293661110360917894, 6.31027458480504666287269709440, 6.31290557489062687614342156901, 6.90956719701582355528530522475, 6.97267993423042363951242290168, 7.01797988008727037178015339679, 7.65052726775847318673802253791, 8.059837860544257229439088438684
