| L(s) = 1 | − 5-s − 2.37·7-s − 4·11-s − 6.74·13-s + 0.372·17-s + 4·19-s − 23-s + 25-s − 0.372·29-s − 2.37·31-s + 2.37·35-s + 0.372·37-s − 9.11·41-s − 4·43-s − 4.74·47-s − 1.37·49-s + 4.37·53-s + 4·55-s − 14.3·59-s − 2·61-s + 6.74·65-s + 9.62·67-s + 2.37·71-s − 1.25·73-s + 9.48·77-s + 9.48·79-s − 15.8·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.896·7-s − 1.20·11-s − 1.87·13-s + 0.0902·17-s + 0.917·19-s − 0.208·23-s + 0.200·25-s − 0.0691·29-s − 0.426·31-s + 0.400·35-s + 0.0612·37-s − 1.42·41-s − 0.609·43-s − 0.692·47-s − 0.196·49-s + 0.600·53-s + 0.539·55-s − 1.87·59-s − 0.256·61-s + 0.836·65-s + 1.17·67-s + 0.281·71-s − 0.146·73-s + 1.08·77-s + 1.06·79-s − 1.74·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4769175729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4769175729\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 2.37T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 6.74T + 13T^{2} \) |
| 17 | \( 1 - 0.372T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 29 | \( 1 + 0.372T + 29T^{2} \) |
| 31 | \( 1 + 2.37T + 31T^{2} \) |
| 37 | \( 1 - 0.372T + 37T^{2} \) |
| 41 | \( 1 + 9.11T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 4.74T + 47T^{2} \) |
| 53 | \( 1 - 4.37T + 53T^{2} \) |
| 59 | \( 1 + 14.3T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 9.62T + 67T^{2} \) |
| 71 | \( 1 - 2.37T + 71T^{2} \) |
| 73 | \( 1 + 1.25T + 73T^{2} \) |
| 79 | \( 1 - 9.48T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 + 6.74T + 89T^{2} \) |
| 97 | \( 1 + 7.48T + 97T^{2} \) |
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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
−7.78528995257728834276143300479, −7.13185210646970560630039252648, −6.63659154652459136049439924999, −5.52096834867068834183701973024, −5.12167129017756136471920067878, −4.34407450425872949378822075948, −3.23733522493577617190370047874, −2.88501937440715585415500182619, −1.88060025084583719953162132411, −0.31743108799320733658050039418,
0.31743108799320733658050039418, 1.88060025084583719953162132411, 2.88501937440715585415500182619, 3.23733522493577617190370047874, 4.34407450425872949378822075948, 5.12167129017756136471920067878, 5.52096834867068834183701973024, 6.63659154652459136049439924999, 7.13185210646970560630039252648, 7.78528995257728834276143300479
