| L(s) = 1 | + 1.11·2-s − 3-s − 0.765·4-s − 1.11·6-s + 0.814·7-s − 3.07·8-s + 9-s + 1.47·11-s + 0.765·12-s − 1.41·13-s + 0.904·14-s − 1.88·16-s − 7.97·17-s + 1.11·18-s − 0.789·19-s − 0.814·21-s + 1.64·22-s − 7.47·23-s + 3.07·24-s − 1.57·26-s − 27-s − 0.623·28-s − 6.33·29-s + 10.4·31-s + 4.05·32-s − 1.47·33-s − 8.86·34-s + ⋯ |
| L(s) = 1 | + 0.785·2-s − 0.577·3-s − 0.382·4-s − 0.453·6-s + 0.307·7-s − 1.08·8-s + 0.333·9-s + 0.445·11-s + 0.221·12-s − 0.392·13-s + 0.241·14-s − 0.470·16-s − 1.93·17-s + 0.261·18-s − 0.181·19-s − 0.177·21-s + 0.350·22-s − 1.55·23-s + 0.627·24-s − 0.308·26-s − 0.192·27-s − 0.117·28-s − 1.17·29-s + 1.88·31-s + 0.716·32-s − 0.257·33-s − 1.52·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.283168933\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.283168933\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 107 | \( 1 - T \) |
| good | 2 | \( 1 - 1.11T + 2T^{2} \) |
| 7 | \( 1 - 0.814T + 7T^{2} \) |
| 11 | \( 1 - 1.47T + 11T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 + 7.97T + 17T^{2} \) |
| 19 | \( 1 + 0.789T + 19T^{2} \) |
| 23 | \( 1 + 7.47T + 23T^{2} \) |
| 29 | \( 1 + 6.33T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 - 1.40T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 - 7.23T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 - 1.85T + 53T^{2} \) |
| 59 | \( 1 + 6.27T + 59T^{2} \) |
| 61 | \( 1 - 8.00T + 61T^{2} \) |
| 67 | \( 1 - 7.68T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 + 1.00T + 79T^{2} \) |
| 83 | \( 1 - 8.35T + 83T^{2} \) |
| 89 | \( 1 + 7.78T + 89T^{2} \) |
| 97 | \( 1 - 6.58T + 97T^{2} \) |
| 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
−7.88675742284288813521731176911, −6.81471465112860094923504588863, −6.29951507952795813956603370739, −5.74882264398092852409174821685, −4.86680232991750699552736684227, −4.33434730416027051809412604036, −3.93196848521818382439071364465, −2.72445624445941355850399006519, −1.91271817198778481521345357915, −0.49722238026425814567641926919,
0.49722238026425814567641926919, 1.91271817198778481521345357915, 2.72445624445941355850399006519, 3.93196848521818382439071364465, 4.33434730416027051809412604036, 4.86680232991750699552736684227, 5.74882264398092852409174821685, 6.29951507952795813956603370739, 6.81471465112860094923504588863, 7.88675742284288813521731176911
