| L(s) = 1 | + 2.56·2-s + 3-s + 4.58·4-s + 1.31·5-s + 2.56·6-s − 7-s + 6.64·8-s + 9-s + 3.37·10-s − 4.88·11-s + 4.58·12-s − 1.82·13-s − 2.56·14-s + 1.31·15-s + 7.88·16-s + 1.68·17-s + 2.56·18-s + 1.82·19-s + 6.02·20-s − 21-s − 12.5·22-s + 3.03·23-s + 6.64·24-s − 3.27·25-s − 4.68·26-s + 27-s − 4.58·28-s + ⋯ |
| L(s) = 1 | + 1.81·2-s + 0.577·3-s + 2.29·4-s + 0.587·5-s + 1.04·6-s − 0.377·7-s + 2.34·8-s + 0.333·9-s + 1.06·10-s − 1.47·11-s + 1.32·12-s − 0.506·13-s − 0.686·14-s + 0.338·15-s + 1.97·16-s + 0.409·17-s + 0.605·18-s + 0.418·19-s + 1.34·20-s − 0.218·21-s − 2.67·22-s + 0.633·23-s + 1.35·24-s − 0.655·25-s − 0.918·26-s + 0.192·27-s − 0.867·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.368253638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.368253638\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 2.56T + 2T^{2} \) |
| 5 | \( 1 - 1.31T + 5T^{2} \) |
| 11 | \( 1 + 4.88T + 11T^{2} \) |
| 13 | \( 1 + 1.82T + 13T^{2} \) |
| 17 | \( 1 - 1.68T + 17T^{2} \) |
| 19 | \( 1 - 1.82T + 19T^{2} \) |
| 23 | \( 1 - 3.03T + 23T^{2} \) |
| 29 | \( 1 + 2.73T + 29T^{2} \) |
| 31 | \( 1 - 2.17T + 31T^{2} \) |
| 37 | \( 1 + 3.96T + 37T^{2} \) |
| 43 | \( 1 + 7.49T + 43T^{2} \) |
| 47 | \( 1 + 2.68T + 47T^{2} \) |
| 53 | \( 1 + 5.35T + 53T^{2} \) |
| 59 | \( 1 - 4.82T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 - 4.06T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 + 4.33T + 73T^{2} \) |
| 79 | \( 1 - 6.58T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 + 8.01T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−10.25743170157145608068547128797, −9.615930864757637304648397103143, −8.189312495533446901770223751414, −7.33517194075607888631575035893, −6.50782936113037012082819177996, −5.41886285188989269743582497024, −5.01454903565453778647537504618, −3.69541855710659112314431525792, −2.87841601699720885220813404395, −2.05180645406019896597074699431,
2.05180645406019896597074699431, 2.87841601699720885220813404395, 3.69541855710659112314431525792, 5.01454903565453778647537504618, 5.41886285188989269743582497024, 6.50782936113037012082819177996, 7.33517194075607888631575035893, 8.189312495533446901770223751414, 9.615930864757637304648397103143, 10.25743170157145608068547128797
