| L(s) = 1 | + 2.16·2-s + 3.21·3-s + 2.66·4-s + 6.94·6-s + 3.61·7-s + 1.44·8-s + 7.34·9-s − 0.578·11-s + 8.58·12-s + 0.0217·13-s + 7.81·14-s − 2.21·16-s − 2.06·17-s + 15.8·18-s − 0.138·19-s + 11.6·21-s − 1.25·22-s − 2.99·23-s + 4.64·24-s + 0.0469·26-s + 13.9·27-s + 9.65·28-s + 4.13·29-s − 6.96·31-s − 7.67·32-s − 1.86·33-s − 4.47·34-s + ⋯ |
| L(s) = 1 | + 1.52·2-s + 1.85·3-s + 1.33·4-s + 2.83·6-s + 1.36·7-s + 0.510·8-s + 2.44·9-s − 0.174·11-s + 2.47·12-s + 0.00603·13-s + 2.08·14-s − 0.554·16-s − 0.501·17-s + 3.73·18-s − 0.0317·19-s + 2.53·21-s − 0.266·22-s − 0.624·23-s + 0.948·24-s + 0.00921·26-s + 2.68·27-s + 1.82·28-s + 0.768·29-s − 1.25·31-s − 1.35·32-s − 0.323·33-s − 0.766·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.88442491\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.88442491\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 2.16T + 2T^{2} \) |
| 3 | \( 1 - 3.21T + 3T^{2} \) |
| 7 | \( 1 - 3.61T + 7T^{2} \) |
| 11 | \( 1 + 0.578T + 11T^{2} \) |
| 13 | \( 1 - 0.0217T + 13T^{2} \) |
| 17 | \( 1 + 2.06T + 17T^{2} \) |
| 19 | \( 1 + 0.138T + 19T^{2} \) |
| 23 | \( 1 + 2.99T + 23T^{2} \) |
| 29 | \( 1 - 4.13T + 29T^{2} \) |
| 31 | \( 1 + 6.96T + 31T^{2} \) |
| 37 | \( 1 - 4.49T + 37T^{2} \) |
| 41 | \( 1 - 0.567T + 41T^{2} \) |
| 43 | \( 1 - 7.53T + 43T^{2} \) |
| 47 | \( 1 + 6.31T + 47T^{2} \) |
| 53 | \( 1 - 0.975T + 53T^{2} \) |
| 59 | \( 1 + 5.85T + 59T^{2} \) |
| 61 | \( 1 - 5.49T + 61T^{2} \) |
| 67 | \( 1 + 6.05T + 67T^{2} \) |
| 71 | \( 1 - 3.12T + 71T^{2} \) |
| 73 | \( 1 - 0.613T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−7.971637322949828718005548429633, −7.46808567745614346727135712736, −6.69986366954972574919426728517, −5.73254469833571428579576065811, −4.83571455177976848390847802869, −4.32832756103305549956691170071, −3.75732659012327971958502474818, −2.87268060369546604042669320309, −2.25587107974681679809671516905, −1.54466535965843884668676345784,
1.54466535965843884668676345784, 2.25587107974681679809671516905, 2.87268060369546604042669320309, 3.75732659012327971958502474818, 4.32832756103305549956691170071, 4.83571455177976848390847802869, 5.73254469833571428579576065811, 6.69986366954972574919426728517, 7.46808567745614346727135712736, 7.971637322949828718005548429633
