L(s) = 1 | + 2-s − 4-s − 5-s + 7-s − 3·8-s − 10-s + 5·13-s + 14-s − 16-s + 7·17-s − 6·19-s + 20-s + 4·23-s − 4·25-s + 5·26-s − 28-s − 9·29-s − 2·31-s + 5·32-s + 7·34-s − 35-s + 9·37-s − 6·38-s + 3·40-s + 7·41-s − 6·43-s + 4·46-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s + 0.377·7-s − 1.06·8-s − 0.316·10-s + 1.38·13-s + 0.267·14-s − 1/4·16-s + 1.69·17-s − 1.37·19-s + 0.223·20-s + 0.834·23-s − 4/5·25-s + 0.980·26-s − 0.188·28-s − 1.67·29-s − 0.359·31-s + 0.883·32-s + 1.20·34-s − 0.169·35-s + 1.47·37-s − 0.973·38-s + 0.474·40-s + 1.09·41-s − 0.914·43-s + 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.195090343\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.195090343\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−7.924542437429132948347184150017, −7.25461409171001655086026789468, −6.00951301896000859750911899657, −5.92767608270396503684304442532, −4.99515890785368532494232706062, −4.19904675844402256642721198697, −3.70766736164884112751416686431, −3.05948199405770213339884044020, −1.78639229445410984565034470782, −0.68303465491592391562695088684,
0.68303465491592391562695088684, 1.78639229445410984565034470782, 3.05948199405770213339884044020, 3.70766736164884112751416686431, 4.19904675844402256642721198697, 4.99515890785368532494232706062, 5.92767608270396503684304442532, 6.00951301896000859750911899657, 7.25461409171001655086026789468, 7.924542437429132948347184150017