| L(s) = 1 | + 1.64·2-s + 0.880·3-s + 1.69·4-s + 0.101·5-s + 1.44·6-s − 7-s + 1.14·8-s − 0.224·9-s + 0.166·10-s + 1.37·11-s + 1.49·12-s + 0.501·13-s − 1.64·14-s + 0.0892·15-s + 0.177·16-s − 0.368·18-s + 0.171·20-s − 0.880·21-s + 2.26·22-s + 1.05·23-s + 1.00·24-s − 0.989·25-s + 0.822·26-s − 1.07·27-s − 1.69·28-s + 0.302·29-s + 0.146·30-s + ⋯ |
| L(s) = 1 | + 1.64·2-s + 0.880·3-s + 1.69·4-s + 0.101·5-s + 1.44·6-s − 7-s + 1.14·8-s − 0.224·9-s + 0.166·10-s + 1.37·11-s + 1.49·12-s + 0.501·13-s − 1.64·14-s + 0.0892·15-s + 0.177·16-s − 0.368·18-s + 0.171·20-s − 0.880·21-s + 2.26·22-s + 1.05·23-s + 1.00·24-s − 0.989·25-s + 0.822·26-s − 1.07·27-s − 1.69·28-s + 0.302·29-s + 0.146·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.649343033\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.649343033\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 353 | \( 1 + T \) |
| good | 2 | \( 1 - 1.64T + T^{2} \) |
| 3 | \( 1 - 0.880T + T^{2} \) |
| 5 | \( 1 - 0.101T + T^{2} \) |
| 11 | \( 1 - 1.37T + T^{2} \) |
| 13 | \( 1 - 0.501T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 1.05T + T^{2} \) |
| 29 | \( 1 - 0.302T + T^{2} \) |
| 31 | \( 1 + 1.83T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.90T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 1.98T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.22T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.135785455471160872384632221438, −8.429643660926593951396234913059, −7.19945599537433372384930281445, −6.64380622241154113955408190710, −5.92708894207106477139244772112, −5.19304229067919705980578785561, −3.88694491342318901548663290380, −3.65083873176218932124497796615, −2.85588723640223790835072638866, −1.81777708751864879487400830128,
1.81777708751864879487400830128, 2.85588723640223790835072638866, 3.65083873176218932124497796615, 3.88694491342318901548663290380, 5.19304229067919705980578785561, 5.92708894207106477139244772112, 6.64380622241154113955408190710, 7.19945599537433372384930281445, 8.429643660926593951396234913059, 9.135785455471160872384632221438
