| L(s) = 1 | + 0.356·2-s − 3-s − 1.87·4-s − 5-s − 0.356·6-s − 7-s − 1.37·8-s + 9-s − 0.356·10-s + 1.09·11-s + 1.87·12-s + 3.51·13-s − 0.356·14-s + 15-s + 3.25·16-s − 2·17-s + 0.356·18-s − 1.23·19-s + 1.87·20-s + 21-s + 0.388·22-s − 23-s + 1.37·24-s + 25-s + 1.25·26-s − 27-s + 1.87·28-s + ⋯ |
| L(s) = 1 | + 0.251·2-s − 0.577·3-s − 0.936·4-s − 0.447·5-s − 0.145·6-s − 0.377·7-s − 0.487·8-s + 0.333·9-s − 0.112·10-s + 0.329·11-s + 0.540·12-s + 0.975·13-s − 0.0951·14-s + 0.258·15-s + 0.813·16-s − 0.485·17-s + 0.0839·18-s − 0.282·19-s + 0.418·20-s + 0.218·21-s + 0.0829·22-s − 0.208·23-s + 0.281·24-s + 0.200·25-s + 0.245·26-s − 0.192·27-s + 0.353·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 0.356T + 2T^{2} \) |
| 11 | \( 1 - 1.09T + 11T^{2} \) |
| 13 | \( 1 - 3.51T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 1.23T + 19T^{2} \) |
| 29 | \( 1 - 0.264T + 29T^{2} \) |
| 31 | \( 1 - 8.90T + 31T^{2} \) |
| 37 | \( 1 + 4.49T + 37T^{2} \) |
| 41 | \( 1 + 1.21T + 41T^{2} \) |
| 43 | \( 1 - 3.39T + 43T^{2} \) |
| 47 | \( 1 + 6.60T + 47T^{2} \) |
| 53 | \( 1 - 4.83T + 53T^{2} \) |
| 59 | \( 1 - 4.56T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 + 9.21T + 67T^{2} \) |
| 71 | \( 1 + 1.48T + 71T^{2} \) |
| 73 | \( 1 - 6.90T + 73T^{2} \) |
| 79 | \( 1 + 0.677T + 79T^{2} \) |
| 83 | \( 1 + 8.08T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 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
−8.594210304155589055287081544971, −7.959823839237487556150994484868, −6.80094568386895177679206936156, −6.20812854801116979407887416171, −5.39055077144511106073610136117, −4.44768989975733312255157843852, −3.93886791221454767665517261645, −2.96029668558119732394383342685, −1.25328890269064510998493944388, 0,
1.25328890269064510998493944388, 2.96029668558119732394383342685, 3.93886791221454767665517261645, 4.44768989975733312255157843852, 5.39055077144511106073610136117, 6.20812854801116979407887416171, 6.80094568386895177679206936156, 7.959823839237487556150994484868, 8.594210304155589055287081544971
