| L(s) = 1 | − 3i·3-s + 4i·5-s − 7-s − 6·9-s − 5i·13-s + 12·15-s − 5·17-s − i·19-s + 3i·21-s + 3·23-s − 11·25-s + 9i·27-s + 7i·29-s − 10·31-s − 4i·35-s + ⋯ |
| L(s) = 1 | − 1.73i·3-s + 1.78i·5-s − 0.377·7-s − 2·9-s − 1.38i·13-s + 3.09·15-s − 1.21·17-s − 0.229i·19-s + 0.654i·21-s + 0.625·23-s − 2.20·25-s + 1.73i·27-s + 1.29i·29-s − 1.79·31-s − 0.676i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + iT \) |
| good | 3 | \( 1 + 3iT - 3T^{2} \) |
| 5 | \( 1 - 4iT - 5T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 5iT - 13T^{2} \) |
| 17 | \( 1 + 5T + 17T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 - 7iT - 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 9iT - 53T^{2} \) |
| 59 | \( 1 + iT - 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 + 7iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 + 14iT - 83T^{2} \) |
| 89 | \( 1 + 4T + 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
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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
−9.078914580800521084834807858492, −8.054932625377401236969486030192, −7.37759537552068899213166869092, −6.75490869545100678888110266108, −6.33333972766039216673932136650, −5.31229610751105295105052134212, −3.34014982079073240885196077070, −2.83619747990884525794359393850, −1.77849389118508217197657670653, 0,
1.97657079152034106338366274028, 3.68374437429460524000993398476, 4.28806962137891571388993652914, 4.94020677998479479840763424810, 5.64627389990694439814382002943, 6.81893578599937410538296245717, 8.364989284097332836323452357892, 8.789738150830262684436493570506, 9.560423801687851858035516854593
