| L(s) = 1 | − 2-s − 0.116·3-s + 4-s − 0.498·5-s + 0.116·6-s − 2.29·7-s − 8-s − 2.98·9-s + 0.498·10-s + 11-s − 0.116·12-s + 0.281·13-s + 2.29·14-s + 0.0579·15-s + 16-s + 1.65·17-s + 2.98·18-s − 0.498·20-s + 0.266·21-s − 22-s + 8.04·23-s + 0.116·24-s − 4.75·25-s − 0.281·26-s + 0.695·27-s − 2.29·28-s − 2.32·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.0670·3-s + 0.5·4-s − 0.222·5-s + 0.0474·6-s − 0.866·7-s − 0.353·8-s − 0.995·9-s + 0.157·10-s + 0.301·11-s − 0.0335·12-s + 0.0779·13-s + 0.612·14-s + 0.0149·15-s + 0.250·16-s + 0.401·17-s + 0.703·18-s − 0.111·20-s + 0.0580·21-s − 0.213·22-s + 1.67·23-s + 0.0237·24-s − 0.950·25-s − 0.0551·26-s + 0.133·27-s − 0.433·28-s − 0.431·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 0.116T + 3T^{2} \) |
| 5 | \( 1 + 0.498T + 5T^{2} \) |
| 7 | \( 1 + 2.29T + 7T^{2} \) |
| 13 | \( 1 - 0.281T + 13T^{2} \) |
| 17 | \( 1 - 1.65T + 17T^{2} \) |
| 23 | \( 1 - 8.04T + 23T^{2} \) |
| 29 | \( 1 + 2.32T + 29T^{2} \) |
| 31 | \( 1 + 3.69T + 31T^{2} \) |
| 37 | \( 1 - 5.88T + 37T^{2} \) |
| 41 | \( 1 - 0.535T + 41T^{2} \) |
| 43 | \( 1 + 5.13T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 - 8.71T + 53T^{2} \) |
| 59 | \( 1 + 2.84T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 - 5.50T + 71T^{2} \) |
| 73 | \( 1 - 2.38T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 - 6.41T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−7.48917002636076467711504826189, −6.96238631846026768293476384466, −6.07197008560984642000583107100, −5.72187244977926378655098532886, −4.71534718775541572619550517286, −3.60880398081215268140900536942, −3.10543629790685987427021589317, −2.23753611502530162228554079713, −1.01244243922355360204530863509, 0,
1.01244243922355360204530863509, 2.23753611502530162228554079713, 3.10543629790685987427021589317, 3.60880398081215268140900536942, 4.71534718775541572619550517286, 5.72187244977926378655098532886, 6.07197008560984642000583107100, 6.96238631846026768293476384466, 7.48917002636076467711504826189
