| L(s) = 1 | + i·2-s + i·3-s − 4-s − 6-s + 2i·7-s − i·8-s − 9-s + 5·11-s − i·12-s − 7i·13-s − 2·14-s + 16-s + i·17-s − i·18-s − 7·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s + 0.755i·7-s − 0.353i·8-s − 0.333·9-s + 1.50·11-s − 0.288i·12-s − 1.94i·13-s − 0.534·14-s + 0.250·16-s + 0.242i·17-s − 0.235i·18-s − 1.60·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.794257456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.794257456\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 + 7iT - 13T^{2} \) |
| 17 | \( 1 - iT - 17T^{2} \) |
| 19 | \( 1 + 7T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 10iT - 43T^{2} \) |
| 47 | \( 1 - iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 - 3iT - 67T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 - 11T + 79T^{2} \) |
| 83 | \( 1 - 7iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 3iT - 97T^{2} \) |
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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
−8.571953254792302678377816762671, −8.026808706453169038851124769377, −6.95725406274032170258570284294, −6.28233620071987847815830609353, −5.66891035323322665365820231056, −5.00860674182862157803170269914, −4.07752429718136462959015246227, −3.42213333691355391738219975985, −2.38214746787555956184166719890, −0.924852083909544195582925709201,
0.63785487248879949804093940345, 1.66631931271587328523285977769, 2.30026517465223915806586651932, 3.58624275696812945686764014702, 4.28188043116021837953331416330, 4.69732987205292821089138934132, 6.27640237502598148488890436183, 6.55539833546994716370610032018, 7.18792545273519804779984190389, 8.282007094233605143303422302981
