| L(s) = 1 | + 3·3-s + 5·5-s + 2·7-s − 18·9-s + 16·11-s − 47·13-s + 15·15-s − 24·17-s + 56·19-s + 6·21-s + 23·23-s + 25·25-s − 135·27-s + 85·29-s − 67·31-s + 48·33-s + 10·35-s + 104·37-s − 141·39-s − 53·41-s + 234·43-s − 90·45-s − 285·47-s − 339·49-s − 72·51-s + 2·53-s + 80·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.107·7-s − 2/3·9-s + 0.438·11-s − 1.00·13-s + 0.258·15-s − 0.342·17-s + 0.676·19-s + 0.0623·21-s + 0.208·23-s + 1/5·25-s − 0.962·27-s + 0.544·29-s − 0.388·31-s + 0.253·33-s + 0.0482·35-s + 0.462·37-s − 0.578·39-s − 0.201·41-s + 0.829·43-s − 0.298·45-s − 0.884·47-s − 0.988·49-s − 0.197·51-s + 0.00518·53-s + 0.196·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 23 | \( 1 - p T \) |
| good | 3 | \( 1 - p T + p^{3} T^{2} \) |
| 7 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 - 16 T + p^{3} T^{2} \) |
| 13 | \( 1 + 47 T + p^{3} T^{2} \) |
| 17 | \( 1 + 24 T + p^{3} T^{2} \) |
| 19 | \( 1 - 56 T + p^{3} T^{2} \) |
| 29 | \( 1 - 85 T + p^{3} T^{2} \) |
| 31 | \( 1 + 67 T + p^{3} T^{2} \) |
| 37 | \( 1 - 104 T + p^{3} T^{2} \) |
| 41 | \( 1 + 53 T + p^{3} T^{2} \) |
| 43 | \( 1 - 234 T + p^{3} T^{2} \) |
| 47 | \( 1 + 285 T + p^{3} T^{2} \) |
| 53 | \( 1 - 2 T + p^{3} T^{2} \) |
| 59 | \( 1 + 80 T + p^{3} T^{2} \) |
| 61 | \( 1 + 764 T + p^{3} T^{2} \) |
| 67 | \( 1 + 236 T + p^{3} T^{2} \) |
| 71 | \( 1 - 289 T + p^{3} T^{2} \) |
| 73 | \( 1 + 225 T + p^{3} T^{2} \) |
| 79 | \( 1 + 24 T + p^{3} T^{2} \) |
| 83 | \( 1 + 684 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1370 T + p^{3} T^{2} \) |
| 97 | \( 1 + 110 T + p^{3} T^{2} \) |
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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
−8.546241752656881089822429056050, −7.78796593542614565394595740067, −6.98700195429505717897886628489, −6.08201204391672153635578831869, −5.24816911562611482315936567781, −4.38071826976140807379768595914, −3.19841085895996877771240800550, −2.52086605439258235145162014515, −1.45074854160159597466073294501, 0,
1.45074854160159597466073294501, 2.52086605439258235145162014515, 3.19841085895996877771240800550, 4.38071826976140807379768595914, 5.24816911562611482315936567781, 6.08201204391672153635578831869, 6.98700195429505717897886628489, 7.78796593542614565394595740067, 8.546241752656881089822429056050
