| L(s) = 1 | − 3·3-s + 4·7-s + 9·9-s + 28·11-s + 16·13-s − 108·17-s − 32·19-s − 12·21-s − 28·23-s − 27·27-s − 238·29-s + 180·31-s − 84·33-s + 40·37-s − 48·39-s + 422·41-s + 276·43-s + 60·47-s − 327·49-s + 324·51-s − 220·53-s + 96·57-s + 804·59-s − 358·61-s + 36·63-s − 884·67-s + 84·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.215·7-s + 1/3·9-s + 0.767·11-s + 0.341·13-s − 1.54·17-s − 0.386·19-s − 0.124·21-s − 0.253·23-s − 0.192·27-s − 1.52·29-s + 1.04·31-s − 0.443·33-s + 0.177·37-s − 0.197·39-s + 1.60·41-s + 0.978·43-s + 0.186·47-s − 0.953·49-s + 0.889·51-s − 0.570·53-s + 0.223·57-s + 1.77·59-s − 0.751·61-s + 0.0719·63-s − 1.61·67-s + 0.146·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 28 T + p^{3} T^{2} \) |
| 13 | \( 1 - 16 T + p^{3} T^{2} \) |
| 17 | \( 1 + 108 T + p^{3} T^{2} \) |
| 19 | \( 1 + 32 T + p^{3} T^{2} \) |
| 23 | \( 1 + 28 T + p^{3} T^{2} \) |
| 29 | \( 1 + 238 T + p^{3} T^{2} \) |
| 31 | \( 1 - 180 T + p^{3} T^{2} \) |
| 37 | \( 1 - 40 T + p^{3} T^{2} \) |
| 41 | \( 1 - 422 T + p^{3} T^{2} \) |
| 43 | \( 1 - 276 T + p^{3} T^{2} \) |
| 47 | \( 1 - 60 T + p^{3} T^{2} \) |
| 53 | \( 1 + 220 T + p^{3} T^{2} \) |
| 59 | \( 1 - 804 T + p^{3} T^{2} \) |
| 61 | \( 1 + 358 T + p^{3} T^{2} \) |
| 67 | \( 1 + 884 T + p^{3} T^{2} \) |
| 71 | \( 1 - 64 T + p^{3} T^{2} \) |
| 73 | \( 1 - 152 T + p^{3} T^{2} \) |
| 79 | \( 1 - 932 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1292 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1146 T + p^{3} T^{2} \) |
| 97 | \( 1 + 824 T + p^{3} T^{2} \) |
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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
−9.056824501490420615404578234727, −8.175954818608658078692231950290, −7.17220429317291622521846894832, −6.39819074001847198406143414584, −5.71449876438814729919219810731, −4.51997523911692224695205998191, −3.95760764225428875690544499542, −2.45726621227577301503835087702, −1.32207985977188144389738280558, 0,
1.32207985977188144389738280558, 2.45726621227577301503835087702, 3.95760764225428875690544499542, 4.51997523911692224695205998191, 5.71449876438814729919219810731, 6.39819074001847198406143414584, 7.17220429317291622521846894832, 8.175954818608658078692231950290, 9.056824501490420615404578234727
