| L(s) = 1 | − 1.58·2-s − 0.358·3-s + 0.525·4-s + 5-s + 0.569·6-s + 2.77·7-s + 2.34·8-s − 2.87·9-s − 1.58·10-s + 11-s − 0.188·12-s + 1.54·13-s − 4.40·14-s − 0.358·15-s − 4.77·16-s + 3.82·17-s + 4.56·18-s + 5.21·19-s + 0.525·20-s − 0.993·21-s − 1.58·22-s + 4.87·23-s − 0.839·24-s + 25-s − 2.45·26-s + 2.10·27-s + 1.45·28-s + ⋯ |
| L(s) = 1 | − 1.12·2-s − 0.206·3-s + 0.262·4-s + 0.447·5-s + 0.232·6-s + 1.04·7-s + 0.828·8-s − 0.957·9-s − 0.502·10-s + 0.301·11-s − 0.0543·12-s + 0.428·13-s − 1.17·14-s − 0.0925·15-s − 1.19·16-s + 0.926·17-s + 1.07·18-s + 1.19·19-s + 0.117·20-s − 0.216·21-s − 0.338·22-s + 1.01·23-s − 0.171·24-s + 0.200·25-s − 0.481·26-s + 0.404·27-s + 0.275·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.220011384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.220011384\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 + 1.58T + 2T^{2} \) |
| 3 | \( 1 + 0.358T + 3T^{2} \) |
| 7 | \( 1 - 2.77T + 7T^{2} \) |
| 13 | \( 1 - 1.54T + 13T^{2} \) |
| 17 | \( 1 - 3.82T + 17T^{2} \) |
| 19 | \( 1 - 5.21T + 19T^{2} \) |
| 23 | \( 1 - 4.87T + 23T^{2} \) |
| 29 | \( 1 + 6.06T + 29T^{2} \) |
| 31 | \( 1 - 2.13T + 31T^{2} \) |
| 37 | \( 1 - 2.69T + 37T^{2} \) |
| 41 | \( 1 + 2.64T + 41T^{2} \) |
| 43 | \( 1 - 2.93T + 43T^{2} \) |
| 47 | \( 1 - 13.5T + 47T^{2} \) |
| 53 | \( 1 + 0.687T + 53T^{2} \) |
| 59 | \( 1 + 2.29T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + 6.87T + 67T^{2} \) |
| 71 | \( 1 - 15.4T + 71T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 3.83T + 83T^{2} \) |
| 89 | \( 1 + 1.55T + 89T^{2} \) |
| 97 | \( 1 - 6.46T + 97T^{2} \) |
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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
−8.521508125530951859580375908656, −7.80384117986060243751433509648, −7.36876703670767337243237834133, −6.27270204250110363300827303903, −5.41033577610081734659599367841, −4.95517680173709015523215916258, −3.80110947809252491703810947091, −2.70423466491848019221969468897, −1.54375358264754763489106463518, −0.843532633409597324824027982773,
0.843532633409597324824027982773, 1.54375358264754763489106463518, 2.70423466491848019221969468897, 3.80110947809252491703810947091, 4.95517680173709015523215916258, 5.41033577610081734659599367841, 6.27270204250110363300827303903, 7.36876703670767337243237834133, 7.80384117986060243751433509648, 8.521508125530951859580375908656
