| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 6·11-s + 2·13-s + 14-s + 16-s − 2·17-s + 5·19-s − 6·22-s + 23-s − 2·26-s − 28-s − 10·29-s − 10·31-s − 32-s + 2·34-s + 2·37-s − 5·38-s − 5·41-s − 8·43-s + 6·44-s − 46-s + 12·47-s − 6·49-s + 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 1.80·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s + 1.14·19-s − 1.27·22-s + 0.208·23-s − 0.392·26-s − 0.188·28-s − 1.85·29-s − 1.79·31-s − 0.176·32-s + 0.342·34-s + 0.328·37-s − 0.811·38-s − 0.780·41-s − 1.21·43-s + 0.904·44-s − 0.147·46-s + 1.75·47-s − 6/7·49-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.35031041781605, −13.91857558995436, −13.25597194323369, −12.74820173510427, −12.32334459733531, −11.50513196421837, −11.29169528493098, −11.06884928948829, −10.02254639195915, −9.749830984910540, −9.205064486845050, −8.869233738377556, −8.399552986678705, −7.519562305166135, −7.184809478287054, −6.684717111308126, −6.137318682547050, −5.579014005565479, −4.986873971502028, −3.913927888742468, −3.724246413452403, −3.138290073080046, −2.093843533100759, −1.600026017875415, −0.9439864067824667, 0,
0.9439864067824667, 1.600026017875415, 2.093843533100759, 3.138290073080046, 3.724246413452403, 3.913927888742468, 4.986873971502028, 5.579014005565479, 6.137318682547050, 6.684717111308126, 7.184809478287054, 7.519562305166135, 8.399552986678705, 8.869233738377556, 9.205064486845050, 9.749830984910540, 10.02254639195915, 11.06884928948829, 11.29169528493098, 11.50513196421837, 12.32334459733531, 12.74820173510427, 13.25597194323369, 13.91857558995436, 14.35031041781605
