| L(s) = 1 | − 4.09e3·2-s + 1.67e7·4-s + 1.37e8·5-s − 3.04e10·7-s − 6.87e10·8-s − 5.61e11·10-s − 2.58e12·11-s − 9.57e13·13-s + 1.24e14·14-s + 2.81e14·16-s + 1.64e15·17-s + 4.95e15·19-s + 2.29e15·20-s + 1.05e16·22-s + 1.07e16·23-s − 2.79e17·25-s + 3.92e17·26-s − 5.10e17·28-s + 1.36e18·29-s − 4.41e18·31-s − 1.15e18·32-s − 6.74e18·34-s − 4.16e18·35-s + 1.01e19·37-s − 2.02e19·38-s − 9.41e18·40-s − 1.58e20·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.251·5-s − 0.830·7-s − 0.353·8-s − 0.177·10-s − 0.248·11-s − 1.13·13-s + 0.587·14-s + 0.250·16-s + 0.685·17-s + 0.513·19-s + 0.125·20-s + 0.175·22-s + 0.102·23-s − 0.936·25-s + 0.805·26-s − 0.415·28-s + 0.717·29-s − 1.00·31-s − 0.176·32-s − 0.484·34-s − 0.208·35-s + 0.254·37-s − 0.362·38-s − 0.0887·40-s − 1.09·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(1.007250771\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.007250771\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4.09e3T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1.37e8T + 2.98e17T^{2} \) |
| 7 | \( 1 + 3.04e10T + 1.34e21T^{2} \) |
| 11 | \( 1 + 2.58e12T + 1.08e26T^{2} \) |
| 13 | \( 1 + 9.57e13T + 7.05e27T^{2} \) |
| 17 | \( 1 - 1.64e15T + 5.77e30T^{2} \) |
| 19 | \( 1 - 4.95e15T + 9.30e31T^{2} \) |
| 23 | \( 1 - 1.07e16T + 1.10e34T^{2} \) |
| 29 | \( 1 - 1.36e18T + 3.63e36T^{2} \) |
| 31 | \( 1 + 4.41e18T + 1.92e37T^{2} \) |
| 37 | \( 1 - 1.01e19T + 1.60e39T^{2} \) |
| 41 | \( 1 + 1.58e20T + 2.08e40T^{2} \) |
| 43 | \( 1 - 1.83e20T + 6.86e40T^{2} \) |
| 47 | \( 1 + 1.40e21T + 6.34e41T^{2} \) |
| 53 | \( 1 - 1.99e21T + 1.27e43T^{2} \) |
| 59 | \( 1 - 4.16e21T + 1.86e44T^{2} \) |
| 61 | \( 1 - 3.42e22T + 4.29e44T^{2} \) |
| 67 | \( 1 - 8.67e22T + 4.48e45T^{2} \) |
| 71 | \( 1 - 5.13e22T + 1.91e46T^{2} \) |
| 73 | \( 1 - 3.49e22T + 3.82e46T^{2} \) |
| 79 | \( 1 - 2.91e23T + 2.75e47T^{2} \) |
| 83 | \( 1 - 1.64e24T + 9.48e47T^{2} \) |
| 89 | \( 1 + 8.74e23T + 5.42e48T^{2} \) |
| 97 | \( 1 - 1.00e25T + 4.66e49T^{2} \) |
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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
−12.96154789154516291696789308609, −11.76747852508582952478656255176, −10.16480568573335293048442514129, −9.462021871151880350361058659983, −7.894258507363011768835935329992, −6.70512306033698209449863210976, −5.30424348819325161259777202821, −3.35808716527520320718013534778, −2.10503367000393694134246411881, −0.56008546852882636750370691766,
0.56008546852882636750370691766, 2.10503367000393694134246411881, 3.35808716527520320718013534778, 5.30424348819325161259777202821, 6.70512306033698209449863210976, 7.894258507363011768835935329992, 9.462021871151880350361058659983, 10.16480568573335293048442514129, 11.76747852508582952478656255176, 12.96154789154516291696789308609
