| L(s) = 1 | + 2.39·2-s + 2.59·3-s + 3.74·4-s + 6.22·6-s + 2.89·7-s + 4.19·8-s + 3.73·9-s − 11-s + 9.72·12-s − 4.73·13-s + 6.93·14-s + 2.56·16-s + 5.65·17-s + 8.94·18-s + 19-s + 7.50·21-s − 2.39·22-s + 4.00·23-s + 10.8·24-s − 11.3·26-s + 1.89·27-s + 10.8·28-s + 9.32·29-s − 6.60·31-s − 2.24·32-s − 2.59·33-s + 13.5·34-s + ⋯ |
| L(s) = 1 | + 1.69·2-s + 1.49·3-s + 1.87·4-s + 2.53·6-s + 1.09·7-s + 1.48·8-s + 1.24·9-s − 0.301·11-s + 2.80·12-s − 1.31·13-s + 1.85·14-s + 0.640·16-s + 1.37·17-s + 2.10·18-s + 0.229·19-s + 1.63·21-s − 0.511·22-s + 0.835·23-s + 2.22·24-s − 2.22·26-s + 0.364·27-s + 2.05·28-s + 1.73·29-s − 1.18·31-s − 0.397·32-s − 0.451·33-s + 2.32·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.42815971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.42815971\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.39T + 2T^{2} \) |
| 3 | \( 1 - 2.59T + 3T^{2} \) |
| 7 | \( 1 - 2.89T + 7T^{2} \) |
| 13 | \( 1 + 4.73T + 13T^{2} \) |
| 17 | \( 1 - 5.65T + 17T^{2} \) |
| 23 | \( 1 - 4.00T + 23T^{2} \) |
| 29 | \( 1 - 9.32T + 29T^{2} \) |
| 31 | \( 1 + 6.60T + 31T^{2} \) |
| 37 | \( 1 + 6.07T + 37T^{2} \) |
| 41 | \( 1 - 5.47T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 - 0.295T + 47T^{2} \) |
| 53 | \( 1 - 3.81T + 53T^{2} \) |
| 59 | \( 1 + 5.54T + 59T^{2} \) |
| 61 | \( 1 + 1.01T + 61T^{2} \) |
| 67 | \( 1 + 6.98T + 67T^{2} \) |
| 71 | \( 1 + 1.02T + 71T^{2} \) |
| 73 | \( 1 - 0.202T + 73T^{2} \) |
| 79 | \( 1 - 7.28T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 + 4.81T + 97T^{2} \) |
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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
−7.996512150494119901654696473393, −7.43574461733056969807525221316, −6.89928140921634323473122548618, −5.70160308885606147837495970503, −4.98641940132444261464107191974, −4.62328194264813800495104990636, −3.56614508186414688229564416770, −3.04332408356760486705739761049, −2.35300601581367010116665016500, −1.52393642980524284788488872558,
1.52393642980524284788488872558, 2.35300601581367010116665016500, 3.04332408356760486705739761049, 3.56614508186414688229564416770, 4.62328194264813800495104990636, 4.98641940132444261464107191974, 5.70160308885606147837495970503, 6.89928140921634323473122548618, 7.43574461733056969807525221316, 7.996512150494119901654696473393
