| L(s) = 1 | − i·2-s − 3-s − 4-s − i·5-s + i·6-s − 7-s + i·8-s + 9-s − 10-s − 11-s + 12-s + i·13-s + i·14-s + i·15-s + 16-s + 5i·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577·3-s − 0.5·4-s − 0.447i·5-s + 0.408i·6-s − 0.377·7-s + 0.353i·8-s + 0.333·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s + 0.277i·13-s + 0.267i·14-s + 0.258i·15-s + 0.250·16-s + 1.21i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9259428630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9259428630\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 + iT \) |
| 37 | \( 1 + (-6 + i)T \) |
| good | 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 - 5iT - 17T^{2} \) |
| 19 | \( 1 - 3iT - 19T^{2} \) |
| 23 | \( 1 - 3iT - 23T^{2} \) |
| 29 | \( 1 + 10iT - 29T^{2} \) |
| 31 | \( 1 - 2iT - 31T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 - 11T + 53T^{2} \) |
| 59 | \( 1 - 10iT - 59T^{2} \) |
| 61 | \( 1 - 14iT - 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + T + 73T^{2} \) |
| 79 | \( 1 + 10iT - 79T^{2} \) |
| 83 | \( 1 + 3T + 83T^{2} \) |
| 89 | \( 1 + 9iT - 89T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
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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
−10.02592590252124287993361174379, −9.262152701564750930883455918575, −8.303894586725900257113948758249, −7.52301015595131319283184415607, −6.19429555039558421062594892004, −5.66328319385818833104268411075, −4.46116264552764782255529051814, −3.78866922353619549736702948144, −2.38677319813039334606711147299, −1.11520569429418792601494087728,
0.52536425682019317544191150121, 2.56023766661685018531833096121, 3.73114384413749356116416006896, 4.96389623615838547188196016628, 5.51968528325553463368885747218, 6.72400727434630523911025153399, 6.99416720499669562100549753595, 8.020486131892934550098755552225, 9.023074740307522681850076441493, 9.757674592322395651360084876713
