| L(s) = 1 | − 2.15e6·3-s + 4.86e9·5-s − 2.14e10·7-s − 2.98e12·9-s + 6.85e13·11-s − 5.86e14·13-s − 1.04e16·15-s − 2.42e16·17-s − 2.43e17·19-s + 4.62e16·21-s − 1.25e17·23-s + 1.62e19·25-s + 2.28e19·27-s − 1.25e19·29-s + 6.23e19·31-s − 1.47e20·33-s − 1.04e20·35-s + 3.67e20·37-s + 1.26e21·39-s + 3.60e21·41-s − 1.19e22·43-s − 1.45e22·45-s − 8.26e21·47-s − 6.52e22·49-s + 5.23e22·51-s + 1.60e23·53-s + 3.33e23·55-s + ⋯ |
| L(s) = 1 | − 0.780·3-s + 1.78·5-s − 0.0837·7-s − 0.391·9-s + 0.598·11-s − 0.537·13-s − 1.39·15-s − 0.594·17-s − 1.32·19-s + 0.0653·21-s − 0.0519·23-s + 2.17·25-s + 1.08·27-s − 0.227·29-s + 0.458·31-s − 0.467·33-s − 0.149·35-s + 0.248·37-s + 0.419·39-s + 0.609·41-s − 1.06·43-s − 0.697·45-s − 0.220·47-s − 0.992·49-s + 0.464·51-s + 0.846·53-s + 1.06·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(14)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{29}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 2.15e6T + 7.62e12T^{2} \) |
| 5 | \( 1 - 4.86e9T + 7.45e18T^{2} \) |
| 7 | \( 1 + 2.14e10T + 6.57e22T^{2} \) |
| 11 | \( 1 - 6.85e13T + 1.31e28T^{2} \) |
| 13 | \( 1 + 5.86e14T + 1.19e30T^{2} \) |
| 17 | \( 1 + 2.42e16T + 1.66e33T^{2} \) |
| 19 | \( 1 + 2.43e17T + 3.36e34T^{2} \) |
| 23 | \( 1 + 1.25e17T + 5.84e36T^{2} \) |
| 29 | \( 1 + 1.25e19T + 3.05e39T^{2} \) |
| 31 | \( 1 - 6.23e19T + 1.84e40T^{2} \) |
| 37 | \( 1 - 3.67e20T + 2.19e42T^{2} \) |
| 41 | \( 1 - 3.60e21T + 3.50e43T^{2} \) |
| 43 | \( 1 + 1.19e22T + 1.26e44T^{2} \) |
| 47 | \( 1 + 8.26e21T + 1.40e45T^{2} \) |
| 53 | \( 1 - 1.60e23T + 3.59e46T^{2} \) |
| 59 | \( 1 + 1.09e24T + 6.50e47T^{2} \) |
| 61 | \( 1 + 1.32e24T + 1.59e48T^{2} \) |
| 67 | \( 1 + 6.27e24T + 2.01e49T^{2} \) |
| 71 | \( 1 - 1.65e25T + 9.63e49T^{2} \) |
| 73 | \( 1 + 7.93e24T + 2.04e50T^{2} \) |
| 79 | \( 1 + 1.61e25T + 1.72e51T^{2} \) |
| 83 | \( 1 + 1.00e26T + 6.53e51T^{2} \) |
| 89 | \( 1 + 1.72e26T + 4.30e52T^{2} \) |
| 97 | \( 1 + 1.03e27T + 4.39e53T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.52104615290299155280654742575, −11.05329488214768924124843278767, −9.944345959183639649049576047529, −8.808678805966874035475858653968, −6.59817148578998491440115772529, −5.91623673508132515856828743659, −4.72042739279402285827651574816, −2.61799227694409812018826469353, −1.51220891613028011845661210947, 0,
1.51220891613028011845661210947, 2.61799227694409812018826469353, 4.72042739279402285827651574816, 5.91623673508132515856828743659, 6.59817148578998491440115772529, 8.808678805966874035475858653968, 9.944345959183639649049576047529, 11.05329488214768924124843278767, 12.52104615290299155280654742575
