| L(s) = 1 | + 2.98·2-s − 3·3-s + 0.921·4-s − 8.96·6-s + 6.37·7-s − 21.1·8-s + 9·9-s + 11·11-s − 2.76·12-s − 46.0·13-s + 19.0·14-s − 70.5·16-s + 117.·17-s + 26.8·18-s − 82.9·19-s − 19.1·21-s + 32.8·22-s + 23.1·23-s + 63.4·24-s − 137.·26-s − 27·27-s + 5.87·28-s + 218.·29-s − 39.0·31-s − 41.4·32-s − 33·33-s + 350.·34-s + ⋯ |
| L(s) = 1 | + 1.05·2-s − 0.577·3-s + 0.115·4-s − 0.609·6-s + 0.344·7-s − 0.934·8-s + 0.333·9-s + 0.301·11-s − 0.0664·12-s − 0.983·13-s + 0.363·14-s − 1.10·16-s + 1.67·17-s + 0.352·18-s − 1.00·19-s − 0.198·21-s + 0.318·22-s + 0.209·23-s + 0.539·24-s − 1.03·26-s − 0.192·27-s + 0.0396·28-s + 1.40·29-s − 0.226·31-s − 0.229·32-s − 0.174·33-s + 1.76·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.437834890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.437834890\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 2.98T + 8T^{2} \) |
| 7 | \( 1 - 6.37T + 343T^{2} \) |
| 13 | \( 1 + 46.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 117.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 82.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 23.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 218.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 39.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 277.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 311.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 406.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 515.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 177.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 18.3T + 2.05e5T^{2} \) |
| 61 | \( 1 - 766.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 100.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.02T + 3.57e5T^{2} \) |
| 73 | \( 1 - 585.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 658.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.25e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.12e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 527.T + 9.12e5T^{2} \) |
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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
−9.955597914304515529913211318834, −9.072980003776775145321985082603, −8.013595366789296955437368567346, −7.00235477899586819578840074844, −6.05348919020574500924056408098, −5.28333178724088461690465183529, −4.57144096482969193403212795215, −3.64190995975904155991099956560, −2.42759242311781226301073823522, −0.76507525392763101201314404358,
0.76507525392763101201314404358, 2.42759242311781226301073823522, 3.64190995975904155991099956560, 4.57144096482969193403212795215, 5.28333178724088461690465183529, 6.05348919020574500924056408098, 7.00235477899586819578840074844, 8.013595366789296955437368567346, 9.072980003776775145321985082603, 9.955597914304515529913211318834
