| L(s) = 1 | + 5·2-s + 17·4-s − 7·7-s + 45·8-s − 12·11-s − 30·13-s − 35·14-s + 89·16-s − 134·17-s − 92·19-s − 60·22-s + 112·23-s − 150·26-s − 119·28-s + 58·29-s − 224·31-s + 85·32-s − 670·34-s + 146·37-s − 460·38-s − 18·41-s − 340·43-s − 204·44-s + 560·46-s + 208·47-s + 49·49-s − 510·52-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 17/8·4-s − 0.377·7-s + 1.98·8-s − 0.328·11-s − 0.640·13-s − 0.668·14-s + 1.39·16-s − 1.91·17-s − 1.11·19-s − 0.581·22-s + 1.01·23-s − 1.13·26-s − 0.803·28-s + 0.371·29-s − 1.29·31-s + 0.469·32-s − 3.37·34-s + 0.648·37-s − 1.96·38-s − 0.0685·41-s − 1.20·43-s − 0.698·44-s + 1.79·46-s + 0.645·47-s + 1/7·49-s − 1.36·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 2 | \( 1 - 5 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 30 T + p^{3} T^{2} \) |
| 17 | \( 1 + 134 T + p^{3} T^{2} \) |
| 19 | \( 1 + 92 T + p^{3} T^{2} \) |
| 23 | \( 1 - 112 T + p^{3} T^{2} \) |
| 29 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 31 | \( 1 + 224 T + p^{3} T^{2} \) |
| 37 | \( 1 - 146 T + p^{3} T^{2} \) |
| 41 | \( 1 + 18 T + p^{3} T^{2} \) |
| 43 | \( 1 + 340 T + p^{3} T^{2} \) |
| 47 | \( 1 - 208 T + p^{3} T^{2} \) |
| 53 | \( 1 + 754 T + p^{3} T^{2} \) |
| 59 | \( 1 + 380 T + p^{3} T^{2} \) |
| 61 | \( 1 - 718 T + p^{3} T^{2} \) |
| 67 | \( 1 + 412 T + p^{3} T^{2} \) |
| 71 | \( 1 - 960 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1066 T + p^{3} T^{2} \) |
| 79 | \( 1 - 896 T + p^{3} T^{2} \) |
| 83 | \( 1 - 436 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1038 T + p^{3} T^{2} \) |
| 97 | \( 1 - 702 T + p^{3} T^{2} \) |
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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
−8.630922165644295098716854860057, −7.44690819838335379183793504096, −6.68391687414205998156653311125, −6.20174460147457692092310364651, −5.07154146795750940103689489110, −4.58773506801867108176623064663, −3.67196439188189314431589556933, −2.68828652888043393862619792301, −1.97790511512788001758041752325, 0,
1.97790511512788001758041752325, 2.68828652888043393862619792301, 3.67196439188189314431589556933, 4.58773506801867108176623064663, 5.07154146795750940103689489110, 6.20174460147457692092310364651, 6.68391687414205998156653311125, 7.44690819838335379183793504096, 8.630922165644295098716854860057
