L(s) = 1 | + 3-s − 2·5-s − 4·7-s + 9-s + 4·11-s − 2·15-s + 2·17-s − 4·21-s − 25-s + 27-s − 10·29-s + 4·31-s + 4·33-s + 8·35-s + 2·37-s − 6·41-s + 12·43-s − 2·45-s + 9·49-s + 2·51-s + 6·53-s − 8·55-s + 12·59-s − 2·61-s − 4·63-s − 8·67-s − 2·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.516·15-s + 0.485·17-s − 0.872·21-s − 1/5·25-s + 0.192·27-s − 1.85·29-s + 0.718·31-s + 0.696·33-s + 1.35·35-s + 0.328·37-s − 0.937·41-s + 1.82·43-s − 0.298·45-s + 9/7·49-s + 0.280·51-s + 0.824·53-s − 1.07·55-s + 1.56·59-s − 0.256·61-s − 0.503·63-s − 0.977·67-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.37235382040457036689100634678, −6.96649990186684717405021617382, −6.18601200095276560790327263263, −5.53809016423786926047852193663, −4.22355958609103164781006933555, −3.84865224713128863415407536303, −3.28094818784894083659485408370, −2.41631776747875790476201462503, −1.17788762168993240747322563898, 0,
1.17788762168993240747322563898, 2.41631776747875790476201462503, 3.28094818784894083659485408370, 3.84865224713128863415407536303, 4.22355958609103164781006933555, 5.53809016423786926047852193663, 6.18601200095276560790327263263, 6.96649990186684717405021617382, 7.37235382040457036689100634678