| L(s) = 1 | + 2.49·2-s − 2.89·3-s + 4.22·4-s − 7.21·6-s − 0.185·7-s + 5.54·8-s + 5.37·9-s + 5.68·11-s − 12.2·12-s + 6.29·13-s − 0.463·14-s + 5.39·16-s + 1.03·17-s + 13.4·18-s − 0.418·19-s + 0.537·21-s + 14.1·22-s + 5.21·23-s − 16.0·24-s + 15.7·26-s − 6.86·27-s − 0.784·28-s − 7.29·29-s − 8.47·31-s + 2.35·32-s − 16.4·33-s + 2.57·34-s + ⋯ |
| L(s) = 1 | + 1.76·2-s − 1.67·3-s + 2.11·4-s − 2.94·6-s − 0.0701·7-s + 1.96·8-s + 1.79·9-s + 1.71·11-s − 3.52·12-s + 1.74·13-s − 0.123·14-s + 1.34·16-s + 0.250·17-s + 3.15·18-s − 0.0959·19-s + 0.117·21-s + 3.02·22-s + 1.08·23-s − 3.27·24-s + 3.07·26-s − 1.32·27-s − 0.148·28-s − 1.35·29-s − 1.52·31-s + 0.417·32-s − 2.86·33-s + 0.442·34-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
| \(L(1)\) |
\(\approx\) |
\(4.418413501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.418413501\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.49T + 2T^{2} \) |
| 3 | \( 1 + 2.89T + 3T^{2} \) |
| 7 | \( 1 + 0.185T + 7T^{2} \) |
| 11 | \( 1 - 5.68T + 11T^{2} \) |
| 13 | \( 1 - 6.29T + 13T^{2} \) |
| 17 | \( 1 - 1.03T + 17T^{2} \) |
| 19 | \( 1 + 0.418T + 19T^{2} \) |
| 23 | \( 1 - 5.21T + 23T^{2} \) |
| 29 | \( 1 + 7.29T + 29T^{2} \) |
| 31 | \( 1 + 8.47T + 31T^{2} \) |
| 37 | \( 1 - 3.20T + 37T^{2} \) |
| 41 | \( 1 - 9.21T + 41T^{2} \) |
| 43 | \( 1 + 1.97T + 43T^{2} \) |
| 47 | \( 1 - 5.03T + 47T^{2} \) |
| 53 | \( 1 - 9.52T + 53T^{2} \) |
| 59 | \( 1 + 4.69T + 59T^{2} \) |
| 61 | \( 1 + 7.67T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 4.48T + 71T^{2} \) |
| 73 | \( 1 + 8.34T + 73T^{2} \) |
| 79 | \( 1 - 5.40T + 79T^{2} \) |
| 83 | \( 1 - 8.45T + 83T^{2} \) |
| 89 | \( 1 + 17.3T + 89T^{2} \) |
| 97 | \( 1 - 1.97T + 97T^{2} \) |
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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
−7.51796831886913022799805163255, −6.80645443820498989208351073397, −6.39172763154733398486129424196, −5.73802055483667530737482082969, −5.45031000337460549278249964795, −4.38846005618016536458706726565, −3.95447142685163425501129904193, −3.31432730874440190493884569443, −1.77445699547555627840476049049, −1.00953863694665219487717111175,
1.00953863694665219487717111175, 1.77445699547555627840476049049, 3.31432730874440190493884569443, 3.95447142685163425501129904193, 4.38846005618016536458706726565, 5.45031000337460549278249964795, 5.73802055483667530737482082969, 6.39172763154733398486129424196, 6.80645443820498989208351073397, 7.51796831886913022799805163255
