| L(s) = 1 | + 1.83·2-s − 0.596·3-s + 1.36·4-s − 1.09·6-s − 2.23·7-s − 1.16·8-s − 2.64·9-s + 11-s − 0.813·12-s + 3.03·13-s − 4.09·14-s − 4.86·16-s − 2.88·17-s − 4.84·18-s + 19-s + 1.33·21-s + 1.83·22-s + 4.68·23-s + 0.696·24-s + 5.56·26-s + 3.36·27-s − 3.04·28-s + 5.80·29-s − 3.86·31-s − 6.59·32-s − 0.596·33-s − 5.28·34-s + ⋯ |
| L(s) = 1 | + 1.29·2-s − 0.344·3-s + 0.681·4-s − 0.446·6-s − 0.844·7-s − 0.412·8-s − 0.881·9-s + 0.301·11-s − 0.234·12-s + 0.840·13-s − 1.09·14-s − 1.21·16-s − 0.699·17-s − 1.14·18-s + 0.229·19-s + 0.290·21-s + 0.391·22-s + 0.976·23-s + 0.142·24-s + 1.09·26-s + 0.648·27-s − 0.575·28-s + 1.07·29-s − 0.694·31-s − 1.16·32-s − 0.103·33-s − 0.906·34-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
| \(L(1)\) |
\(\approx\) |
\(2.390117740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.390117740\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.83T + 2T^{2} \) |
| 3 | \( 1 + 0.596T + 3T^{2} \) |
| 7 | \( 1 + 2.23T + 7T^{2} \) |
| 13 | \( 1 - 3.03T + 13T^{2} \) |
| 17 | \( 1 + 2.88T + 17T^{2} \) |
| 23 | \( 1 - 4.68T + 23T^{2} \) |
| 29 | \( 1 - 5.80T + 29T^{2} \) |
| 31 | \( 1 + 3.86T + 31T^{2} \) |
| 37 | \( 1 + 2.50T + 37T^{2} \) |
| 41 | \( 1 - 2.12T + 41T^{2} \) |
| 43 | \( 1 + 7.08T + 43T^{2} \) |
| 47 | \( 1 - 8.99T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + 1.87T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 2.18T + 67T^{2} \) |
| 71 | \( 1 - 1.25T + 71T^{2} \) |
| 73 | \( 1 - 7.41T + 73T^{2} \) |
| 79 | \( 1 + 3.19T + 79T^{2} \) |
| 83 | \( 1 - 17.3T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + 16.9T + 97T^{2} \) |
| 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
−8.328216767910762703977999638111, −6.97253114830212031241939288744, −6.59582147891656766732518532458, −5.89200507376078946326684116960, −5.34091277736447800814226483077, −4.56747631169079570729383264199, −3.67232223841579221999003284962, −3.16518315842024114880522747255, −2.30401148777546707706170948222, −0.67573457625165018005715487226,
0.67573457625165018005715487226, 2.30401148777546707706170948222, 3.16518315842024114880522747255, 3.67232223841579221999003284962, 4.56747631169079570729383264199, 5.34091277736447800814226483077, 5.89200507376078946326684116960, 6.59582147891656766732518532458, 6.97253114830212031241939288744, 8.328216767910762703977999638111
