| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 2·6-s − 2·9-s − 3·11-s − 2·12-s − 13-s − 4·16-s − 7·17-s − 4·18-s − 6·22-s + 6·23-s − 2·26-s + 5·27-s − 5·29-s − 2·31-s − 8·32-s + 3·33-s − 14·34-s − 4·36-s + 2·37-s + 39-s − 2·41-s − 4·43-s − 6·44-s + 12·46-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s − 2/3·9-s − 0.904·11-s − 0.577·12-s − 0.277·13-s − 16-s − 1.69·17-s − 0.942·18-s − 1.27·22-s + 1.25·23-s − 0.392·26-s + 0.962·27-s − 0.928·29-s − 0.359·31-s − 1.41·32-s + 0.522·33-s − 2.40·34-s − 2/3·36-s + 0.328·37-s + 0.160·39-s − 0.312·41-s − 0.609·43-s − 0.904·44-s + 1.76·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−9.233562639444269397435350431299, −8.562401232917240169463749007291, −7.28758295351193845600460393855, −6.53798490672310802047008418274, −5.66958692619558457468511044947, −5.07649126191872099449172898860, −4.33447975604545784013755045008, −3.14066572792900855376603714204, −2.30839442218736782075025152213, 0,
2.30839442218736782075025152213, 3.14066572792900855376603714204, 4.33447975604545784013755045008, 5.07649126191872099449172898860, 5.66958692619558457468511044947, 6.53798490672310802047008418274, 7.28758295351193845600460393855, 8.562401232917240169463749007291, 9.233562639444269397435350431299
