L(s) = 1 | + 5-s − 3·9-s − 4·11-s + 6·13-s − 2·17-s + 25-s + 6·29-s + 8·31-s − 10·37-s − 2·41-s − 4·43-s − 3·45-s + 8·47-s − 2·53-s − 4·55-s − 8·59-s + 14·61-s + 6·65-s + 12·67-s + 16·71-s − 2·73-s + 8·79-s + 9·81-s + 8·83-s − 2·85-s − 10·89-s − 2·97-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 9-s − 1.20·11-s + 1.66·13-s − 0.485·17-s + 1/5·25-s + 1.11·29-s + 1.43·31-s − 1.64·37-s − 0.312·41-s − 0.609·43-s − 0.447·45-s + 1.16·47-s − 0.274·53-s − 0.539·55-s − 1.04·59-s + 1.79·61-s + 0.744·65-s + 1.46·67-s + 1.89·71-s − 0.234·73-s + 0.900·79-s + 81-s + 0.878·83-s − 0.216·85-s − 1.05·89-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.784452441\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.784452441\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−8.400283535029584108147575940934, −8.058922554914412122379413087558, −6.78248070030119110364171611800, −6.28136292817627051453406742773, −5.47254939185735647334240808646, −4.90054686617114529340998403638, −3.72179923773102648718320811330, −2.92257849808604956447035538813, −2.09663760025842118849257213691, −0.75960958349138602912815032456,
0.75960958349138602912815032456, 2.09663760025842118849257213691, 2.92257849808604956447035538813, 3.72179923773102648718320811330, 4.90054686617114529340998403638, 5.47254939185735647334240808646, 6.28136292817627051453406742773, 6.78248070030119110364171611800, 8.058922554914412122379413087558, 8.400283535029584108147575940934