| L(s) = 1 | − 1.56·2-s + 3-s + 0.445·4-s + 0.640·5-s − 1.56·6-s + 3.30·7-s + 2.43·8-s + 9-s − 1.00·10-s − 4.71·11-s + 0.445·12-s − 13-s − 5.16·14-s + 0.640·15-s − 4.69·16-s + 3.70·17-s − 1.56·18-s + 1.83·19-s + 0.285·20-s + 3.30·21-s + 7.36·22-s − 4.83·23-s + 2.43·24-s − 4.58·25-s + 1.56·26-s + 27-s + 1.47·28-s + ⋯ |
| L(s) = 1 | − 1.10·2-s + 0.577·3-s + 0.222·4-s + 0.286·5-s − 0.638·6-s + 1.24·7-s + 0.859·8-s + 0.333·9-s − 0.316·10-s − 1.42·11-s + 0.128·12-s − 0.277·13-s − 1.38·14-s + 0.165·15-s − 1.17·16-s + 0.897·17-s − 0.368·18-s + 0.421·19-s + 0.0638·20-s + 0.721·21-s + 1.57·22-s − 1.00·23-s + 0.496·24-s − 0.917·25-s + 0.306·26-s + 0.192·27-s + 0.278·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 1.56T + 2T^{2} \) |
| 5 | \( 1 - 0.640T + 5T^{2} \) |
| 7 | \( 1 - 3.30T + 7T^{2} \) |
| 11 | \( 1 + 4.71T + 11T^{2} \) |
| 17 | \( 1 - 3.70T + 17T^{2} \) |
| 19 | \( 1 - 1.83T + 19T^{2} \) |
| 23 | \( 1 + 4.83T + 23T^{2} \) |
| 29 | \( 1 - 3.49T + 29T^{2} \) |
| 31 | \( 1 + 8.89T + 31T^{2} \) |
| 37 | \( 1 + 9.16T + 37T^{2} \) |
| 41 | \( 1 - 3.86T + 41T^{2} \) |
| 43 | \( 1 + 8.55T + 43T^{2} \) |
| 47 | \( 1 - 0.198T + 47T^{2} \) |
| 53 | \( 1 + 2.01T + 53T^{2} \) |
| 59 | \( 1 + 1.76T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 + 2.02T + 71T^{2} \) |
| 73 | \( 1 + 8.31T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 9.67T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−8.132335280271253921293284653861, −7.66274375622368918573019752149, −7.16497320202475902584072903731, −5.70929994837487172081305040011, −5.12021663511615196282878300642, −4.34063788653956608713165700140, −3.21215738245217255420644710359, −2.05013961592564703964815403486, −1.52589377920639088470854515941, 0,
1.52589377920639088470854515941, 2.05013961592564703964815403486, 3.21215738245217255420644710359, 4.34063788653956608713165700140, 5.12021663511615196282878300642, 5.70929994837487172081305040011, 7.16497320202475902584072903731, 7.66274375622368918573019752149, 8.132335280271253921293284653861
