L(s) = 1 | + 2.79i·2-s − 1.15i·3-s − 3.82·4-s + 3.24·6-s − 6.64·7-s + 0.480i·8-s + 7.65·9-s − 6.24·11-s + 4.43i·12-s − 18.6i·13-s − 18.5i·14-s − 16.6·16-s − 29.3·17-s + 21.4i·18-s + (3.89 + 18.5i)19-s + ⋯ |
L(s) = 1 | + 1.39i·2-s − 0.386i·3-s − 0.957·4-s + 0.540·6-s − 0.949·7-s + 0.0600i·8-s + 0.850·9-s − 0.567·11-s + 0.369i·12-s − 1.43i·13-s − 1.32i·14-s − 1.04·16-s − 1.72·17-s + 1.19i·18-s + (0.205 + 0.978i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.205 + 0.978i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.205 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2594768727\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2594768727\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (-3.89 - 18.5i)T \) |
good | 2 | \( 1 - 2.79iT - 4T^{2} \) |
| 3 | \( 1 + 1.15iT - 9T^{2} \) |
| 7 | \( 1 + 6.64T + 49T^{2} \) |
| 11 | \( 1 + 6.24T + 121T^{2} \) |
| 13 | \( 1 + 18.6iT - 169T^{2} \) |
| 17 | \( 1 + 29.3T + 289T^{2} \) |
| 23 | \( 1 + 19.9T + 529T^{2} \) |
| 29 | \( 1 + 29.4iT - 841T^{2} \) |
| 31 | \( 1 + 26.2iT - 961T^{2} \) |
| 37 | \( 1 - 47.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 41.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 14.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 5.50T + 2.20e3T^{2} \) |
| 53 | \( 1 + 55.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 86.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 20.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 0.363iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 100. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 0.472T + 5.32e3T^{2} \) |
| 79 | \( 1 - 120. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 42.1T + 6.88e3T^{2} \) |
| 89 | \( 1 + 4.51iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 11.5iT - 9.40e3T^{2} \) |
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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
−10.30250319169182706289077682941, −9.652550438149398376009241179912, −8.299278825164217175369094674507, −7.82836103977858310730277529999, −6.79151731253890342402531309684, −6.21058089727886748621270441150, −5.23886815393178327290864066844, −3.97015830640740308761687586498, −2.35245850151161886471685521586, −0.098719838110280187201087057147,
1.75120735906533236319529274264, 2.86617409404588863934570840527, 4.04191243156398383499037078850, 4.71097314664842030563519724167, 6.51193551544483362663190692435, 7.13755453438823124616954301540, 9.003634803245273235653959551953, 9.319154879817651748750587295405, 10.30020450480713864717697345323, 10.89422080004862009961873493747