| L(s) = 1 | − 1.96·3-s + 0.807·5-s − 3.34·7-s + 0.848·9-s − 4.04·11-s − 4.44·13-s − 1.58·15-s − 5.91·17-s − 2.98·19-s + 6.55·21-s − 7.45·23-s − 4.34·25-s + 4.22·27-s + 7.98·29-s − 9.58·31-s + 7.93·33-s − 2.69·35-s + 2.67·37-s + 8.71·39-s + 3.85·41-s − 9.57·43-s + 0.685·45-s + 9.46·47-s + 4.17·49-s + 11.6·51-s − 9.89·53-s − 3.26·55-s + ⋯ |
| L(s) = 1 | − 1.13·3-s + 0.361·5-s − 1.26·7-s + 0.282·9-s − 1.21·11-s − 1.23·13-s − 0.408·15-s − 1.43·17-s − 0.684·19-s + 1.43·21-s − 1.55·23-s − 0.869·25-s + 0.812·27-s + 1.48·29-s − 1.72·31-s + 1.38·33-s − 0.456·35-s + 0.440·37-s + 1.39·39-s + 0.601·41-s − 1.46·43-s + 0.102·45-s + 1.38·47-s + 0.596·49-s + 1.62·51-s − 1.35·53-s − 0.440·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01286480563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01286480563\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 + 1.96T + 3T^{2} \) |
| 5 | \( 1 - 0.807T + 5T^{2} \) |
| 7 | \( 1 + 3.34T + 7T^{2} \) |
| 11 | \( 1 + 4.04T + 11T^{2} \) |
| 13 | \( 1 + 4.44T + 13T^{2} \) |
| 17 | \( 1 + 5.91T + 17T^{2} \) |
| 19 | \( 1 + 2.98T + 19T^{2} \) |
| 23 | \( 1 + 7.45T + 23T^{2} \) |
| 29 | \( 1 - 7.98T + 29T^{2} \) |
| 31 | \( 1 + 9.58T + 31T^{2} \) |
| 37 | \( 1 - 2.67T + 37T^{2} \) |
| 41 | \( 1 - 3.85T + 41T^{2} \) |
| 43 | \( 1 + 9.57T + 43T^{2} \) |
| 47 | \( 1 - 9.46T + 47T^{2} \) |
| 53 | \( 1 + 9.89T + 53T^{2} \) |
| 59 | \( 1 + 6.88T + 59T^{2} \) |
| 61 | \( 1 + 7.56T + 61T^{2} \) |
| 67 | \( 1 - 8.97T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 - 3.51T + 73T^{2} \) |
| 79 | \( 1 - 9.64T + 79T^{2} \) |
| 83 | \( 1 + 9.42T + 83T^{2} \) |
| 89 | \( 1 - 0.414T + 89T^{2} \) |
| 97 | \( 1 - 8.39T + 97T^{2} \) |
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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
−8.387553926275176660299772274149, −7.61786558539588270078420376976, −6.62710159528752554205534048099, −6.32457208511920033614983438780, −5.52995698523825272866225685408, −4.87932414431251299362258435499, −4.00810009658666076631985605504, −2.75743034608549314679703088660, −2.12271695362634481158367496367, −0.06240768949309930971462466713,
0.06240768949309930971462466713, 2.12271695362634481158367496367, 2.75743034608549314679703088660, 4.00810009658666076631985605504, 4.87932414431251299362258435499, 5.52995698523825272866225685408, 6.32457208511920033614983438780, 6.62710159528752554205534048099, 7.61786558539588270078420376976, 8.387553926275176660299772274149
