| L(s) = 1 | − 4.47·3-s − 5·5-s + 31.3·7-s − 6.99·9-s + 8.94·11-s − 62·13-s + 22.3·15-s − 46·17-s − 107.·19-s − 140·21-s − 192.·23-s + 25·25-s + 152.·27-s − 90·29-s + 152.·31-s − 40.0·33-s − 156.·35-s − 214·37-s + 277.·39-s − 10·41-s + 67.0·43-s + 34.9·45-s − 398.·47-s + 637.·49-s + 205.·51-s − 678·53-s − 44.7·55-s + ⋯ |
| L(s) = 1 | − 0.860·3-s − 0.447·5-s + 1.69·7-s − 0.259·9-s + 0.245·11-s − 1.32·13-s + 0.384·15-s − 0.656·17-s − 1.29·19-s − 1.45·21-s − 1.74·23-s + 0.200·25-s + 1.08·27-s − 0.576·29-s + 0.880·31-s − 0.211·33-s − 0.755·35-s − 0.950·37-s + 1.13·39-s − 0.0380·41-s + 0.237·43-s + 0.115·45-s − 1.23·47-s + 1.85·49-s + 0.564·51-s − 1.75·53-s − 0.109·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 3 | \( 1 + 4.47T + 27T^{2} \) |
| 7 | \( 1 - 31.3T + 343T^{2} \) |
| 11 | \( 1 - 8.94T + 1.33e3T^{2} \) |
| 13 | \( 1 + 62T + 2.19e3T^{2} \) |
| 17 | \( 1 + 46T + 4.91e3T^{2} \) |
| 19 | \( 1 + 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 192.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 90T + 2.43e4T^{2} \) |
| 31 | \( 1 - 152.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 214T + 5.06e4T^{2} \) |
| 41 | \( 1 + 10T + 6.89e4T^{2} \) |
| 43 | \( 1 - 67.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 398.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 678T + 1.48e5T^{2} \) |
| 59 | \( 1 - 411.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 250T + 2.26e5T^{2} \) |
| 67 | \( 1 + 49.1T + 3.00e5T^{2} \) |
| 71 | \( 1 - 366.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 522T + 3.89e5T^{2} \) |
| 79 | \( 1 + 876.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 380.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 970T + 7.04e5T^{2} \) |
| 97 | \( 1 + 934T + 9.12e5T^{2} \) |
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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
−11.72635719987079053075994375186, −11.19990040512809189244998901798, −10.18354381072050069418901439625, −8.590225515539206919452026639226, −7.80295355920318112079605761779, −6.47073839427840834831047605747, −5.14506212575153012544161861903, −4.34505219403782920223278302616, −2.05225850346205488929694359608, 0,
2.05225850346205488929694359608, 4.34505219403782920223278302616, 5.14506212575153012544161861903, 6.47073839427840834831047605747, 7.80295355920318112079605761779, 8.590225515539206919452026639226, 10.18354381072050069418901439625, 11.19990040512809189244998901798, 11.72635719987079053075994375186
