| L(s) = 1 | − 2.45·2-s − 2.31·3-s + 4.05·4-s − 2.20·5-s + 5.69·6-s − 0.890·7-s − 5.04·8-s + 2.36·9-s + 5.42·10-s − 1.81·11-s − 9.37·12-s + 13-s + 2.19·14-s + 5.10·15-s + 4.30·16-s − 2.96·17-s − 5.80·18-s − 7.45·19-s − 8.93·20-s + 2.06·21-s + 4.45·22-s − 8.54·23-s + 11.6·24-s − 0.136·25-s − 2.45·26-s + 1.47·27-s − 3.60·28-s + ⋯ |
| L(s) = 1 | − 1.73·2-s − 1.33·3-s + 2.02·4-s − 0.986·5-s + 2.32·6-s − 0.336·7-s − 1.78·8-s + 0.786·9-s + 1.71·10-s − 0.546·11-s − 2.70·12-s + 0.277·13-s + 0.585·14-s + 1.31·15-s + 1.07·16-s − 0.718·17-s − 1.36·18-s − 1.71·19-s − 1.99·20-s + 0.449·21-s + 0.950·22-s − 1.78·23-s + 2.38·24-s − 0.0273·25-s − 0.482·26-s + 0.284·27-s − 0.681·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 617 | \( 1 - T \) |
| good | 2 | \( 1 + 2.45T + 2T^{2} \) |
| 3 | \( 1 + 2.31T + 3T^{2} \) |
| 5 | \( 1 + 2.20T + 5T^{2} \) |
| 7 | \( 1 + 0.890T + 7T^{2} \) |
| 11 | \( 1 + 1.81T + 11T^{2} \) |
| 17 | \( 1 + 2.96T + 17T^{2} \) |
| 19 | \( 1 + 7.45T + 19T^{2} \) |
| 23 | \( 1 + 8.54T + 23T^{2} \) |
| 29 | \( 1 - 10.0T + 29T^{2} \) |
| 31 | \( 1 + 7.75T + 31T^{2} \) |
| 37 | \( 1 - 7.12T + 37T^{2} \) |
| 41 | \( 1 - 1.41T + 41T^{2} \) |
| 43 | \( 1 - 4.26T + 43T^{2} \) |
| 47 | \( 1 - 8.74T + 47T^{2} \) |
| 53 | \( 1 - 2.66T + 53T^{2} \) |
| 59 | \( 1 + 4.22T + 59T^{2} \) |
| 61 | \( 1 + 15.1T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 - 15.0T + 71T^{2} \) |
| 73 | \( 1 - 9.95T + 73T^{2} \) |
| 79 | \( 1 + 6.85T + 79T^{2} \) |
| 83 | \( 1 + 5.19T + 83T^{2} \) |
| 89 | \( 1 - 9.77T + 89T^{2} \) |
| 97 | \( 1 - 1.22T + 97T^{2} \) |
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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
−7.71932638280096740099317539205, −6.84801401128942078995564232125, −6.28217179137627562304353926706, −5.86894172862206528057162623570, −4.58804402546936471477599018007, −4.07196917721574182127013671644, −2.73916352304737596498463542305, −1.86616091593988969356726130725, −0.59384274550663543261710394487, 0,
0.59384274550663543261710394487, 1.86616091593988969356726130725, 2.73916352304737596498463542305, 4.07196917721574182127013671644, 4.58804402546936471477599018007, 5.86894172862206528057162623570, 6.28217179137627562304353926706, 6.84801401128942078995564232125, 7.71932638280096740099317539205
