L(s) = 1 | + (1.06 − 1.06i)2-s + (1.36 + 1.06i)3-s − 0.267i·4-s + (1.06 − 1.06i)5-s + (2.58 − 0.320i)6-s + (1 − i)7-s + (1.84 + 1.84i)8-s + (0.732 + 2.90i)9-s − 2.26i·10-s + (−2.90 − 2.90i)11-s + (0.285 − 0.366i)12-s − 2.12i·14-s + (2.58 − 0.320i)15-s + 4.46·16-s − 5.03·17-s + (3.87 + 2.31i)18-s + ⋯ |
L(s) = 1 | + (0.752 − 0.752i)2-s + (0.788 + 0.614i)3-s − 0.133i·4-s + (0.476 − 0.476i)5-s + (1.05 − 0.130i)6-s + (0.377 − 0.377i)7-s + (0.652 + 0.652i)8-s + (0.244 + 0.969i)9-s − 0.717i·10-s + (−0.877 − 0.877i)11-s + (0.0823 − 0.105i)12-s − 0.569i·14-s + (0.668 − 0.0827i)15-s + 1.11·16-s − 1.22·17-s + (0.913 + 0.546i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.86228 - 0.532898i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.86228 - 0.532898i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.36 - 1.06i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-1.06 + 1.06i)T - 2iT^{2} \) |
| 5 | \( 1 + (-1.06 + 1.06i)T - 5iT^{2} \) |
| 7 | \( 1 + (-1 + i)T - 7iT^{2} \) |
| 11 | \( 1 + (2.90 + 2.90i)T + 11iT^{2} \) |
| 17 | \( 1 + 5.03T + 17T^{2} \) |
| 19 | \( 1 + (-2.73 - 2.73i)T + 19iT^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 7.16iT - 29T^{2} \) |
| 31 | \( 1 + (2.46 + 2.46i)T + 31iT^{2} \) |
| 37 | \( 1 + (-3.83 + 3.83i)T - 37iT^{2} \) |
| 41 | \( 1 + (3.97 - 3.97i)T - 41iT^{2} \) |
| 43 | \( 1 + 2.19iT - 43T^{2} \) |
| 47 | \( 1 + (4.25 + 4.25i)T + 47iT^{2} \) |
| 53 | \( 1 + 0.779iT - 53T^{2} \) |
| 59 | \( 1 + (-2.12 - 2.12i)T + 59iT^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 + (4.19 + 4.19i)T + 67iT^{2} \) |
| 71 | \( 1 + (2.12 - 2.12i)T - 71iT^{2} \) |
| 73 | \( 1 + (0.901 - 0.901i)T - 73iT^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 + (-2.90 + 2.90i)T - 83iT^{2} \) |
| 89 | \( 1 + (6.59 + 6.59i)T + 89iT^{2} \) |
| 97 | \( 1 + (1.19 + 1.19i)T + 97iT^{2} \) |
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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
−10.95198414802991696957101292982, −10.15171292485844053296679093461, −9.154318785867621140225729363824, −8.222459905895879400605221690755, −7.57790574212608012525415617709, −5.73911111070681762882942283032, −4.84587586974155783040704100822, −3.99998641039788071664614668989, −2.95619030771835067367110588427, −1.89297381903310559239221867201,
1.81752162278138020617486767639, 2.94254894935334599259644899458, 4.47870776692308441017448149064, 5.37013179320599808497001464728, 6.56723316570405305843951644703, 7.06067127043950770358897344460, 7.998350963894187265899303185961, 9.071323402122738379241839899382, 9.992742874938662075944084549952, 10.89651616997035956432906061162