| L(s) = 1 | + 0.646·3-s + 0.913i·7-s − 2.58·9-s + 3.94i·11-s + (2.44 + 2.64i)13-s − 6.19·17-s − 1.11i·19-s + 0.590i·21-s − 5.54·23-s − 3.60·27-s − 1.58·29-s − 9.60i·31-s + 2.55i·33-s + 7.84i·37-s + (1.58 + 1.70i)39-s + ⋯ |
| L(s) = 1 | + 0.373·3-s + 0.345i·7-s − 0.860·9-s + 1.19i·11-s + (0.679 + 0.733i)13-s − 1.50·17-s − 0.256i·19-s + 0.128i·21-s − 1.15·23-s − 0.694·27-s − 0.293·29-s − 1.72i·31-s + 0.443i·33-s + 1.28i·37-s + (0.253 + 0.273i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 - 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9252337369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9252337369\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-2.44 - 2.64i)T \) |
| good | 3 | \( 1 - 0.646T + 3T^{2} \) |
| 7 | \( 1 - 0.913iT - 7T^{2} \) |
| 11 | \( 1 - 3.94iT - 11T^{2} \) |
| 17 | \( 1 + 6.19T + 17T^{2} \) |
| 19 | \( 1 + 1.11iT - 19T^{2} \) |
| 23 | \( 1 + 5.54T + 23T^{2} \) |
| 29 | \( 1 + 1.58T + 29T^{2} \) |
| 31 | \( 1 + 9.60iT - 31T^{2} \) |
| 37 | \( 1 - 7.84iT - 37T^{2} \) |
| 41 | \( 1 - 5.06iT - 41T^{2} \) |
| 43 | \( 1 + 6.83T + 43T^{2} \) |
| 47 | \( 1 - 9.66iT - 47T^{2} \) |
| 53 | \( 1 - 1.29T + 53T^{2} \) |
| 59 | \( 1 - 9.01iT - 59T^{2} \) |
| 61 | \( 1 + 5.58T + 61T^{2} \) |
| 67 | \( 1 - 7.84iT - 67T^{2} \) |
| 71 | \( 1 + 6.77iT - 71T^{2} \) |
| 73 | \( 1 + 7.84iT - 73T^{2} \) |
| 79 | \( 1 - 7.16T + 79T^{2} \) |
| 83 | \( 1 - 16.5iT - 83T^{2} \) |
| 89 | \( 1 + 11.3iT - 89T^{2} \) |
| 97 | \( 1 - 1.63iT - 97T^{2} \) |
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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
−9.707513637327953583612751269521, −9.154742191767009357506229726188, −8.406776165285893338214551593548, −7.62859911932544302632632069823, −6.56188774366128043973056102589, −5.95994203776742116190246446874, −4.70010199360894094032460953651, −4.00439883971581715161482911503, −2.65567461151976430728930326285, −1.87469368740724184559212813121,
0.33797176891473699201138144220, 2.05014792940737600025871264057, 3.25235216888852368993284052765, 3.89105274087016832816321797959, 5.27063185214434030693766625443, 5.98865311368772004895930817497, 6.84098300966995858509454421705, 7.972353279242934668943480692366, 8.593517775442703392173444896246, 9.043407739432762031350452460198
