| L(s) = 1 | + 2.30·2-s + 1.30·3-s + 3.30·4-s + 3·6-s + 4.30·7-s + 3.00·8-s − 1.30·9-s + 4.30·12-s + 5·13-s + 9.90·14-s + 0.302·16-s − 3.90·17-s − 3.00·18-s + 19-s + 5.60·21-s + 3.69·23-s + 3.90·24-s + 11.5·26-s − 5.60·27-s + 14.2·28-s + 9.90·29-s − 4.21·31-s − 5.30·32-s − 9·34-s − 4.30·36-s − 9.60·37-s + 2.30·38-s + ⋯ |
| L(s) = 1 | + 1.62·2-s + 0.752·3-s + 1.65·4-s + 1.22·6-s + 1.62·7-s + 1.06·8-s − 0.434·9-s + 1.24·12-s + 1.38·13-s + 2.64·14-s + 0.0756·16-s − 0.947·17-s − 0.707·18-s + 0.229·19-s + 1.22·21-s + 0.770·23-s + 0.797·24-s + 2.25·26-s − 1.07·27-s + 2.68·28-s + 1.83·29-s − 0.756·31-s − 0.937·32-s − 1.54·34-s − 0.717·36-s − 1.57·37-s + 0.373·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.243327029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.243327029\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.30T + 2T^{2} \) |
| 3 | \( 1 - 1.30T + 3T^{2} \) |
| 7 | \( 1 - 4.30T + 7T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 + 3.90T + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 3.69T + 23T^{2} \) |
| 29 | \( 1 - 9.90T + 29T^{2} \) |
| 31 | \( 1 + 4.21T + 31T^{2} \) |
| 37 | \( 1 + 9.60T + 37T^{2} \) |
| 41 | \( 1 + 1.60T + 41T^{2} \) |
| 43 | \( 1 + 7.21T + 43T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 + 2.30T + 53T^{2} \) |
| 59 | \( 1 - 0.211T + 59T^{2} \) |
| 61 | \( 1 + 2.90T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 4.60T + 71T^{2} \) |
| 73 | \( 1 - 2.90T + 73T^{2} \) |
| 79 | \( 1 - 0.0916T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 - 5.30T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−8.556937497151398184849386919334, −8.064284100652035077754966358894, −6.99747118106898413266568301272, −6.31119973292927319909044966918, −5.31805249920525050442029239063, −4.89082703020526102414841369392, −3.98415232419867606725209567684, −3.29911630817377591555704907183, −2.37838838754784752685392958090, −1.50741989665557882504566693155,
1.50741989665557882504566693155, 2.37838838754784752685392958090, 3.29911630817377591555704907183, 3.98415232419867606725209567684, 4.89082703020526102414841369392, 5.31805249920525050442029239063, 6.31119973292927319909044966918, 6.99747118106898413266568301272, 8.064284100652035077754966358894, 8.556937497151398184849386919334
