| L(s) = 1 | − 1.49·2-s + 3-s + 0.223·4-s − 1.49·6-s + 1.33·7-s + 2.64·8-s + 9-s + 3.95·11-s + 0.223·12-s + 0.0410·13-s − 1.99·14-s − 4.39·16-s − 6.69·17-s − 1.49·18-s − 5.69·19-s + 1.33·21-s − 5.89·22-s + 4.16·23-s + 2.64·24-s − 0.0612·26-s + 27-s + 0.299·28-s − 7.48·29-s + 0.727·31-s + 1.26·32-s + 3.95·33-s + 9.98·34-s + ⋯ |
| L(s) = 1 | − 1.05·2-s + 0.577·3-s + 0.111·4-s − 0.608·6-s + 0.504·7-s + 0.936·8-s + 0.333·9-s + 1.19·11-s + 0.0646·12-s + 0.0113·13-s − 0.532·14-s − 1.09·16-s − 1.62·17-s − 0.351·18-s − 1.30·19-s + 0.291·21-s − 1.25·22-s + 0.868·23-s + 0.540·24-s − 0.0120·26-s + 0.192·27-s + 0.0565·28-s − 1.38·29-s + 0.130·31-s + 0.222·32-s + 0.688·33-s + 1.71·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 + 1.49T + 2T^{2} \) |
| 7 | \( 1 - 1.33T + 7T^{2} \) |
| 11 | \( 1 - 3.95T + 11T^{2} \) |
| 13 | \( 1 - 0.0410T + 13T^{2} \) |
| 17 | \( 1 + 6.69T + 17T^{2} \) |
| 19 | \( 1 + 5.69T + 19T^{2} \) |
| 23 | \( 1 - 4.16T + 23T^{2} \) |
| 29 | \( 1 + 7.48T + 29T^{2} \) |
| 31 | \( 1 - 0.727T + 31T^{2} \) |
| 37 | \( 1 + 5.03T + 37T^{2} \) |
| 41 | \( 1 + 2.90T + 41T^{2} \) |
| 43 | \( 1 + 5.70T + 43T^{2} \) |
| 53 | \( 1 + 5.79T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 + 4.38T + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 + 15.2T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + 4.68T + 89T^{2} \) |
| 97 | \( 1 - 8.64T + 97T^{2} \) |
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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
−8.457589207726104392791055775565, −7.68478797733414722114282794472, −6.88868690392948628743063897512, −6.34058838013265962933160122836, −4.88355679201532644282541365947, −4.37390285861818777840213953836, −3.47977421843107956093960934880, −2.06387001953641043853225414123, −1.52114944393531946336576512629, 0,
1.52114944393531946336576512629, 2.06387001953641043853225414123, 3.47977421843107956093960934880, 4.37390285861818777840213953836, 4.88355679201532644282541365947, 6.34058838013265962933160122836, 6.88868690392948628743063897512, 7.68478797733414722114282794472, 8.457589207726104392791055775565
