L(s) = 1 | − 2-s + (0.536 + 0.536i)3-s + 4-s + (−2.23 − 0.127i)5-s + (−0.536 − 0.536i)6-s + (0.767 + 0.767i)7-s − 8-s − 2.42i·9-s + (2.23 + 0.127i)10-s + 4.39i·11-s + (0.536 + 0.536i)12-s + 6.74·13-s + (−0.767 − 0.767i)14-s + (−1.12 − 1.26i)15-s + 16-s + 7.34i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.309 + 0.309i)3-s + 0.5·4-s + (−0.998 − 0.0571i)5-s + (−0.219 − 0.219i)6-s + (0.290 + 0.290i)7-s − 0.353·8-s − 0.808i·9-s + (0.705 + 0.0404i)10-s + 1.32i·11-s + (0.154 + 0.154i)12-s + 1.87·13-s + (−0.205 − 0.205i)14-s + (−0.291 − 0.326i)15-s + 0.250·16-s + 1.78i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.802326 + 0.489132i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.802326 + 0.489132i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 + (2.23 + 0.127i)T \) |
| 37 | \( 1 + (-6.04 - 0.633i)T \) |
good | 3 | \( 1 + (-0.536 - 0.536i)T + 3iT^{2} \) |
| 7 | \( 1 + (-0.767 - 0.767i)T + 7iT^{2} \) |
| 11 | \( 1 - 4.39iT - 11T^{2} \) |
| 13 | \( 1 - 6.74T + 13T^{2} \) |
| 17 | \( 1 - 7.34iT - 17T^{2} \) |
| 19 | \( 1 + (2.59 - 2.59i)T - 19iT^{2} \) |
| 23 | \( 1 - 1.20T + 23T^{2} \) |
| 29 | \( 1 + (1.25 + 1.25i)T + 29iT^{2} \) |
| 31 | \( 1 + (4.14 - 4.14i)T - 31iT^{2} \) |
| 41 | \( 1 - 4.07iT - 41T^{2} \) |
| 43 | \( 1 + 8.56T + 43T^{2} \) |
| 47 | \( 1 + (-7.68 - 7.68i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.31 + 2.31i)T - 53iT^{2} \) |
| 59 | \( 1 + (-7.61 + 7.61i)T - 59iT^{2} \) |
| 61 | \( 1 + (-1.14 + 1.14i)T - 61iT^{2} \) |
| 67 | \( 1 + (-6.25 + 6.25i)T - 67iT^{2} \) |
| 71 | \( 1 + 5.46T + 71T^{2} \) |
| 73 | \( 1 + (1.88 + 1.88i)T + 73iT^{2} \) |
| 79 | \( 1 + (5.13 - 5.13i)T - 79iT^{2} \) |
| 83 | \( 1 + (-0.570 + 0.570i)T - 83iT^{2} \) |
| 89 | \( 1 + (7.54 + 7.54i)T + 89iT^{2} \) |
| 97 | \( 1 + 5.39iT - 97T^{2} \) |
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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
−11.36556820883056863407576363104, −10.63540929523805678505360724096, −9.651768416117621174174194272689, −8.486896700602345151557878900724, −8.307635307273956549335690547932, −6.94958425922972055595891682828, −6.01441385892366341713050377685, −4.25182232808081372223240095544, −3.50279409985366115401881304839, −1.57498701703147038738978322923,
0.865287218671018101826048689564, 2.76886333314809316531144963513, 3.96180408119657111913556146283, 5.48312056699927288382531036098, 6.84023544553937522234617642602, 7.65547178197472295655974375457, 8.497724961830795629062060478467, 8.976016077829635454927211228455, 10.63888344079404118892256147635, 11.15942386887266794319881225171