L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s − 3·9-s + 2·10-s − 2·13-s + 16-s + 6·17-s − 3·18-s − 4·19-s + 2·20-s + 8·23-s − 25-s − 2·26-s + 2·29-s + 31-s + 32-s + 6·34-s − 3·36-s + 10·37-s − 4·38-s + 2·40-s + 6·41-s + 8·43-s − 6·45-s + 8·46-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s − 9-s + 0.632·10-s − 0.554·13-s + 1/4·16-s + 1.45·17-s − 0.707·18-s − 0.917·19-s + 0.447·20-s + 1.66·23-s − 1/5·25-s − 0.392·26-s + 0.371·29-s + 0.179·31-s + 0.176·32-s + 1.02·34-s − 1/2·36-s + 1.64·37-s − 0.648·38-s + 0.316·40-s + 0.937·41-s + 1.21·43-s − 0.894·45-s + 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3038 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3038 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.385213602\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.385213602\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 31 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.767717911672973124583558991985, −7.85520408569034144174211600673, −7.13164735275382460400374069877, −6.10481334469888763444585228494, −5.73966246913035569800644487489, −4.99493437014528324856589187903, −4.05486645798658988472696574741, −2.88035578683009317838785569193, −2.44397171990298187609757473706, −1.04640991976554111715719591186,
1.04640991976554111715719591186, 2.44397171990298187609757473706, 2.88035578683009317838785569193, 4.05486645798658988472696574741, 4.99493437014528324856589187903, 5.73966246913035569800644487489, 6.10481334469888763444585228494, 7.13164735275382460400374069877, 7.85520408569034144174211600673, 8.767717911672973124583558991985