| L(s) = 1 | − 3·2-s + 4·4-s + 2·5-s − 2·7-s − 3·8-s − 6·10-s + 4·13-s + 6·14-s + 3·16-s − 4·17-s + 2·19-s + 8·20-s + 8·23-s − 7·25-s − 12·26-s − 8·28-s − 8·29-s + 8·31-s − 6·32-s + 12·34-s − 4·35-s − 2·37-s − 6·38-s − 6·40-s + 8·43-s − 24·46-s + 4·47-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 2·4-s + 0.894·5-s − 0.755·7-s − 1.06·8-s − 1.89·10-s + 1.10·13-s + 1.60·14-s + 3/4·16-s − 0.970·17-s + 0.458·19-s + 1.78·20-s + 1.66·23-s − 7/5·25-s − 2.35·26-s − 1.51·28-s − 1.48·29-s + 1.43·31-s − 1.06·32-s + 2.05·34-s − 0.676·35-s − 0.328·37-s − 0.973·38-s − 0.948·40-s + 1.21·43-s − 3.53·46-s + 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9720589790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9720589790\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 25 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 69 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 55 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 93 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 20 T + 186 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 9 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 118 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 24 T + 285 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 138 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.072537442315104861818779814518, −7.904965801330110946848607183565, −7.47561577045377581762215693499, −7.03812140418263025581557444131, −6.70382492671431974127035999018, −6.66671550612483106125045788763, −5.93101041532867661772299164012, −5.66539276884666793242116601304, −5.61136172808665014301371687997, −5.09058976147054486344967319685, −4.30746572293853628224035602271, −4.08913388499634500596246846280, −3.67371225275942759703907226113, −3.16813340021828198557317052522, −2.61038805662279907323391203982, −2.39998067071151164606138676194, −1.73013334471359669612718285780, −1.41020087113041439000108157070, −0.73781535584866070058013628589, −0.50696692675460602036342334643,
0.50696692675460602036342334643, 0.73781535584866070058013628589, 1.41020087113041439000108157070, 1.73013334471359669612718285780, 2.39998067071151164606138676194, 2.61038805662279907323391203982, 3.16813340021828198557317052522, 3.67371225275942759703907226113, 4.08913388499634500596246846280, 4.30746572293853628224035602271, 5.09058976147054486344967319685, 5.61136172808665014301371687997, 5.66539276884666793242116601304, 5.93101041532867661772299164012, 6.66671550612483106125045788763, 6.70382492671431974127035999018, 7.03812140418263025581557444131, 7.47561577045377581762215693499, 7.904965801330110946848607183565, 8.072537442315104861818779814518
