| L(s) = 1 | + 9.60·3-s − 7.92·5-s − 170.·7-s − 150.·9-s − 106.·11-s − 524.·13-s − 76.2·15-s + 429.·17-s + 1.55e3·19-s − 1.64e3·21-s − 3.35e3·23-s − 3.06e3·25-s − 3.78e3·27-s + 7.79e3·29-s − 4.51e3·31-s − 1.02e3·33-s + 1.35e3·35-s + 7.97e3·37-s − 5.04e3·39-s − 1.68e3·41-s − 2.19e4·43-s + 1.19e3·45-s + 2.07e4·47-s + 1.23e4·49-s + 4.12e3·51-s + 3.57e4·53-s + 845.·55-s + ⋯ |
| L(s) = 1 | + 0.616·3-s − 0.141·5-s − 1.31·7-s − 0.619·9-s − 0.265·11-s − 0.860·13-s − 0.0874·15-s + 0.360·17-s + 0.991·19-s − 0.812·21-s − 1.32·23-s − 0.979·25-s − 0.998·27-s + 1.72·29-s − 0.843·31-s − 0.163·33-s + 0.186·35-s + 0.957·37-s − 0.530·39-s − 0.156·41-s − 1.81·43-s + 0.0879·45-s + 1.36·47-s + 0.735·49-s + 0.222·51-s + 1.74·53-s + 0.0376·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.288280617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.288280617\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 41 | \( 1 + 1.68e3T \) |
| good | 3 | \( 1 - 9.60T + 243T^{2} \) |
| 5 | \( 1 + 7.92T + 3.12e3T^{2} \) |
| 7 | \( 1 + 170.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 106.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 524.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 429.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.55e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.35e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.79e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.51e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.97e3T + 6.93e7T^{2} \) |
| 43 | \( 1 + 2.19e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.07e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.57e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.91e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.61e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.79e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.54e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.60e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.57e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.53e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.03e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.48e5T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.856959448902914494563743454242, −8.946767448101891907246983942120, −8.026735426183860528561723632657, −7.25543753851992017911696229826, −6.18245737501238286200903769550, −5.34376755150844523378068291042, −3.92322992734548199230736534986, −3.08145436196551024431662290199, −2.26905911437521216884328820716, −0.49205469729112657636664117268,
0.49205469729112657636664117268, 2.26905911437521216884328820716, 3.08145436196551024431662290199, 3.92322992734548199230736534986, 5.34376755150844523378068291042, 6.18245737501238286200903769550, 7.25543753851992017911696229826, 8.026735426183860528561723632657, 8.946767448101891907246983942120, 9.856959448902914494563743454242
