| L(s) = 1 | + 2.41·2-s + 2·3-s + 3.82·4-s + 4.82·6-s − 0.828·7-s + 4.41·8-s + 9-s − 4.82·11-s + 7.65·12-s + 2·13-s − 1.99·14-s + 2.99·16-s + 2.82·17-s + 2.41·18-s + 0.828·19-s − 1.65·21-s − 11.6·22-s + 8.82·23-s + 8.82·24-s + 4.82·26-s − 4·27-s − 3.17·28-s + 29-s − 10.4·31-s − 1.58·32-s − 9.65·33-s + 6.82·34-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 1.15·3-s + 1.91·4-s + 1.97·6-s − 0.313·7-s + 1.56·8-s + 0.333·9-s − 1.45·11-s + 2.21·12-s + 0.554·13-s − 0.534·14-s + 0.749·16-s + 0.685·17-s + 0.569·18-s + 0.190·19-s − 0.361·21-s − 2.48·22-s + 1.84·23-s + 1.80·24-s + 0.946·26-s − 0.769·27-s − 0.599·28-s + 0.185·29-s − 1.88·31-s − 0.280·32-s − 1.68·33-s + 1.17·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.093544203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.093544203\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 3 | \( 1 - 2T + 3T^{2} \) |
| 7 | \( 1 + 0.828T + 7T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 - 0.828T + 19T^{2} \) |
| 23 | \( 1 - 8.82T + 23T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 - 0.343T + 47T^{2} \) |
| 53 | \( 1 + 7.65T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 7.65T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 7.31T + 71T^{2} \) |
| 73 | \( 1 - 8.48T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 - 3.65T + 89T^{2} \) |
| 97 | \( 1 + 4.48T + 97T^{2} \) |
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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
−10.70805783205948962967836344103, −9.515027063884336566485376147055, −8.555675792174676392509444852728, −7.61243358840106910512735351582, −6.81438054490549224466571823971, −5.55383582150764953460208867895, −5.01908759001608458643886494178, −3.53377652688439377213213686244, −3.18674292458814768027802117835, −2.12374313181917302432899613908,
2.12374313181917302432899613908, 3.18674292458814768027802117835, 3.53377652688439377213213686244, 5.01908759001608458643886494178, 5.55383582150764953460208867895, 6.81438054490549224466571823971, 7.61243358840106910512735351582, 8.555675792174676392509444852728, 9.515027063884336566485376147055, 10.70805783205948962967836344103
