| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 6·11-s + 2·13-s − 14-s + 16-s + 19-s + 6·22-s + 6·23-s − 5·25-s − 2·26-s + 28-s − 6·29-s + 8·31-s − 32-s − 10·37-s − 38-s + 6·41-s − 4·43-s − 6·44-s − 6·46-s − 6·47-s + 49-s + 5·50-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 + 0.229·19-s + 1.27·22-s + 1.25·23-s − 25-s − 0.392·26-s + 0.188·28-s − 1.11·29-s + 1.43·31-s − 0.176·32-s − 1.64·37-s − 0.162·38-s + 0.937·41-s − 0.609·43-s − 0.904·44-s − 0.884·46-s − 0.875·47-s + 1/7·49-s + 0.707·50-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.490670546431863178338078429995, −7.87462094921647883942664875297, −7.33615990354905827830525466937, −6.31820508463320297458061667389, −5.44565375834713779887264721264, −4.76918027065692334568092858822, −3.42948835695246631266124027247, −2.57033400147183809405890786244, −1.47814962499483450164849405751, 0,
1.47814962499483450164849405751, 2.57033400147183809405890786244, 3.42948835695246631266124027247, 4.76918027065692334568092858822, 5.44565375834713779887264721264, 6.31820508463320297458061667389, 7.33615990354905827830525466937, 7.87462094921647883942664875297, 8.490670546431863178338078429995
