| L(s) = 1 | + 1.53·2-s − 1.36i·3-s + 0.356·4-s − 5-s − 2.09i·6-s + 1.36i·7-s − 2.52·8-s + 1.13·9-s − 1.53·10-s + (−3.14 + 1.05i)11-s − 0.487i·12-s − 5.07·13-s + 2.10i·14-s + 1.36i·15-s − 4.58·16-s − 7.63i·17-s + ⋯ |
| L(s) = 1 | + 1.08·2-s − 0.788i·3-s + 0.178·4-s − 0.447·5-s − 0.856i·6-s + 0.517i·7-s − 0.891·8-s + 0.378·9-s − 0.485·10-s + (−0.947 + 0.319i)11-s − 0.140i·12-s − 1.40·13-s + 0.561i·14-s + 0.352i·15-s − 1.14·16-s − 1.85i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5117793378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5117793378\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 + (3.14 - 1.05i)T \) |
| 19 | \( 1 + (3.99 + 1.75i)T \) |
| good | 2 | \( 1 - 1.53T + 2T^{2} \) |
| 3 | \( 1 + 1.36iT - 3T^{2} \) |
| 7 | \( 1 - 1.36iT - 7T^{2} \) |
| 13 | \( 1 + 5.07T + 13T^{2} \) |
| 17 | \( 1 + 7.63iT - 17T^{2} \) |
| 23 | \( 1 + 1.64T + 23T^{2} \) |
| 29 | \( 1 + 1.47T + 29T^{2} \) |
| 31 | \( 1 - 4.43iT - 31T^{2} \) |
| 37 | \( 1 - 2.89iT - 37T^{2} \) |
| 41 | \( 1 + 4.91T + 41T^{2} \) |
| 43 | \( 1 - 0.286iT - 43T^{2} \) |
| 47 | \( 1 - 4.33T + 47T^{2} \) |
| 53 | \( 1 - 12.4iT - 53T^{2} \) |
| 59 | \( 1 + 7.90iT - 59T^{2} \) |
| 61 | \( 1 - 15.1iT - 61T^{2} \) |
| 67 | \( 1 + 4.88iT - 67T^{2} \) |
| 71 | \( 1 + 7.90iT - 71T^{2} \) |
| 73 | \( 1 + 10.4iT - 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 1.19iT - 83T^{2} \) |
| 89 | \( 1 + 8.37iT - 89T^{2} \) |
| 97 | \( 1 + 13.3iT - 97T^{2} \) |
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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
−9.469024720409533013530319200050, −8.591637301167473880532677884060, −7.42394371171982963909192778163, −7.09076890685702463940814022807, −5.92925761907632930633340538414, −4.90430398088451283667586909231, −4.52666212784085143001397307470, −2.97439358325405112134486380916, −2.30176393015411658447220762789, −0.14833461861488434656399449607,
2.32589008442761693579436246122, 3.72298080413432738742644337135, 4.08302198439148524193945529062, 4.96368912778692703353485363460, 5.74190284612849779514183193446, 6.83376155307386673572458898249, 7.85531374009245698131845013850, 8.643539444465892535587462657292, 9.805902949851798782050166442828, 10.33289740943958237816761979030
