| L(s) = 1 | − 3-s − 0.619·5-s − 1.05·7-s + 9-s + 2.94·11-s − 6.55·13-s + 0.619·15-s + 6.19·17-s − 7.10·19-s + 1.05·21-s − 4.16·23-s − 4.61·25-s − 27-s − 0.996·29-s − 7.31·31-s − 2.94·33-s + 0.655·35-s + 5.89·37-s + 6.55·39-s + 9.96·41-s − 4.82·43-s − 0.619·45-s − 6.61·47-s − 5.87·49-s − 6.19·51-s + 10.9·53-s − 1.82·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.277·5-s − 0.400·7-s + 0.333·9-s + 0.887·11-s − 1.81·13-s + 0.160·15-s + 1.50·17-s − 1.62·19-s + 0.230·21-s − 0.868·23-s − 0.923·25-s − 0.192·27-s − 0.185·29-s − 1.31·31-s − 0.512·33-s + 0.110·35-s + 0.969·37-s + 1.04·39-s + 1.55·41-s − 0.736·43-s − 0.0923·45-s − 0.964·47-s − 0.839·49-s − 0.867·51-s + 1.49·53-s − 0.246·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8382724889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8382724889\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 0.619T + 5T^{2} \) |
| 7 | \( 1 + 1.05T + 7T^{2} \) |
| 11 | \( 1 - 2.94T + 11T^{2} \) |
| 13 | \( 1 + 6.55T + 13T^{2} \) |
| 17 | \( 1 - 6.19T + 17T^{2} \) |
| 19 | \( 1 + 7.10T + 19T^{2} \) |
| 23 | \( 1 + 4.16T + 23T^{2} \) |
| 29 | \( 1 + 0.996T + 29T^{2} \) |
| 31 | \( 1 + 7.31T + 31T^{2} \) |
| 37 | \( 1 - 5.89T + 37T^{2} \) |
| 41 | \( 1 - 9.96T + 41T^{2} \) |
| 43 | \( 1 + 4.82T + 43T^{2} \) |
| 47 | \( 1 + 6.61T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 - 9.89T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 - 3.71T + 79T^{2} \) |
| 83 | \( 1 - 5.90T + 83T^{2} \) |
| 89 | \( 1 - 9.88T + 89T^{2} \) |
| 97 | \( 1 + 8.96T + 97T^{2} \) |
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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
−7.70072574164785821794633267418, −7.15313865058307376800526811295, −6.43058406596906994263988428884, −5.77049867066175534122758139141, −5.12417286223347282505745420730, −4.15502718235871556182599807975, −3.78515477983134906180488393380, −2.58610033244942366837803721436, −1.77307071402811282840477986323, −0.45405927250343831920566649579,
0.45405927250343831920566649579, 1.77307071402811282840477986323, 2.58610033244942366837803721436, 3.78515477983134906180488393380, 4.15502718235871556182599807975, 5.12417286223347282505745420730, 5.77049867066175534122758139141, 6.43058406596906994263988428884, 7.15313865058307376800526811295, 7.70072574164785821794633267418
