| L(s) = 1 | + 4.69·2-s + 14.0·4-s + 4.47·5-s − 27.2·7-s + 28.5·8-s + 21.0·10-s − 5.99·11-s − 127.·14-s + 21.4·16-s − 105.·17-s + 156.·19-s + 62.9·20-s − 28.1·22-s + 175.·23-s − 104.·25-s − 382.·28-s − 204.·29-s − 31.9·31-s − 127.·32-s − 493.·34-s − 121.·35-s − 344.·37-s + 735.·38-s + 127.·40-s + 46.5·41-s − 173.·43-s − 84.3·44-s + ⋯ |
| L(s) = 1 | + 1.66·2-s + 1.75·4-s + 0.400·5-s − 1.46·7-s + 1.26·8-s + 0.664·10-s − 0.164·11-s − 2.44·14-s + 0.335·16-s − 1.49·17-s + 1.88·19-s + 0.703·20-s − 0.272·22-s + 1.59·23-s − 0.839·25-s − 2.58·28-s − 1.31·29-s − 0.185·31-s − 0.703·32-s − 2.48·34-s − 0.587·35-s − 1.52·37-s + 3.13·38-s + 0.504·40-s + 0.177·41-s − 0.614·43-s − 0.288·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 4.69T + 8T^{2} \) |
| 5 | \( 1 - 4.47T + 125T^{2} \) |
| 7 | \( 1 + 27.2T + 343T^{2} \) |
| 11 | \( 1 + 5.99T + 1.33e3T^{2} \) |
| 17 | \( 1 + 105.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 156.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 175.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 204.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 31.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 344.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 46.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 173.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 265.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 172.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 137.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 58.9T + 2.26e5T^{2} \) |
| 67 | \( 1 + 211.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 436.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.15e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.01e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 150.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 565.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 286.T + 9.12e5T^{2} \) |
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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
−8.951807150797317708859091043703, −7.30529383855687195654438332828, −6.88747080498973190164394673025, −6.00523801314290213026860058468, −5.40643369036703236838257455788, −4.52943315625764752602754256770, −3.38233036020837022782639455566, −3.03929746714142521518696885274, −1.82351625270297335094613494892, 0,
1.82351625270297335094613494892, 3.03929746714142521518696885274, 3.38233036020837022782639455566, 4.52943315625764752602754256770, 5.40643369036703236838257455788, 6.00523801314290213026860058468, 6.88747080498973190164394673025, 7.30529383855687195654438332828, 8.951807150797317708859091043703
