L(s) = 1 | + 2.58·2-s − 0.247·3-s + 4.67·4-s − 0.639·6-s + 0.401·7-s + 6.90·8-s − 2.93·9-s + 5.19·11-s − 1.15·12-s + 2.88·13-s + 1.03·14-s + 8.48·16-s + 4.14·17-s − 7.59·18-s − 0.0994·21-s + 13.4·22-s − 3.35·23-s − 1.71·24-s + 7.46·26-s + 1.47·27-s + 1.87·28-s − 6.58·29-s + 6.26·31-s + 8.11·32-s − 1.28·33-s + 10.7·34-s − 13.7·36-s + ⋯ |
L(s) = 1 | + 1.82·2-s − 0.143·3-s + 2.33·4-s − 0.261·6-s + 0.151·7-s + 2.44·8-s − 0.979·9-s + 1.56·11-s − 0.334·12-s + 0.801·13-s + 0.277·14-s + 2.12·16-s + 1.00·17-s − 1.78·18-s − 0.0217·21-s + 2.85·22-s − 0.700·23-s − 0.349·24-s + 1.46·26-s + 0.283·27-s + 0.354·28-s − 1.22·29-s + 1.12·31-s + 1.43·32-s − 0.223·33-s + 1.83·34-s − 2.28·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.707663337\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.707663337\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 - 2.58T + 2T^{2} \) |
| 3 | \( 1 + 0.247T + 3T^{2} \) |
| 7 | \( 1 - 0.401T + 7T^{2} \) |
| 11 | \( 1 - 5.19T + 11T^{2} \) |
| 13 | \( 1 - 2.88T + 13T^{2} \) |
| 17 | \( 1 - 4.14T + 17T^{2} \) |
| 23 | \( 1 + 3.35T + 23T^{2} \) |
| 29 | \( 1 + 6.58T + 29T^{2} \) |
| 31 | \( 1 - 6.26T + 31T^{2} \) |
| 37 | \( 1 - 1.14T + 37T^{2} \) |
| 41 | \( 1 + 2.85T + 41T^{2} \) |
| 43 | \( 1 - 12.4T + 43T^{2} \) |
| 47 | \( 1 + 6.76T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 - 1.51T + 59T^{2} \) |
| 61 | \( 1 + 6.98T + 61T^{2} \) |
| 67 | \( 1 + 0.757T + 67T^{2} \) |
| 71 | \( 1 - 8.33T + 71T^{2} \) |
| 73 | \( 1 - 9.91T + 73T^{2} \) |
| 79 | \( 1 + 3.18T + 79T^{2} \) |
| 83 | \( 1 + 5.51T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 + 8.89T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.50302645203783176682828841346, −6.60501839859655659230625343898, −6.16591372461298631890930304160, −5.68442872083169344343111993149, −5.00018403442333636179315507480, −4.06918756188470873268399516079, −3.70006795873242457230437782947, −2.97442292093439227198345912681, −2.03025807931810821548104361509, −1.09261862133830114643268369867,
1.09261862133830114643268369867, 2.03025807931810821548104361509, 2.97442292093439227198345912681, 3.70006795873242457230437782947, 4.06918756188470873268399516079, 5.00018403442333636179315507480, 5.68442872083169344343111993149, 6.16591372461298631890930304160, 6.60501839859655659230625343898, 7.50302645203783176682828841346