L(s) = 1 | − 2-s + 3.27·3-s + 4-s − 3.27·6-s + 7-s − 8-s + 7.72·9-s + 11-s + 3.27·12-s − 6.46·13-s − 14-s + 16-s − 3.38·17-s − 7.72·18-s + 6.62·19-s + 3.27·21-s − 22-s + 5.53·23-s − 3.27·24-s + 6.46·26-s + 15.4·27-s + 28-s − 1.01·29-s − 2.72·31-s − 32-s + 3.27·33-s + 3.38·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.89·3-s + 0.5·4-s − 1.33·6-s + 0.377·7-s − 0.353·8-s + 2.57·9-s + 0.301·11-s + 0.945·12-s − 1.79·13-s − 0.267·14-s + 0.250·16-s − 0.820·17-s − 1.82·18-s + 1.51·19-s + 0.714·21-s − 0.213·22-s + 1.15·23-s − 0.668·24-s + 1.26·26-s + 2.97·27-s + 0.188·28-s − 0.188·29-s − 0.489·31-s − 0.176·32-s + 0.570·33-s + 0.579·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.047996371\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.047996371\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 - 3.27T + 3T^{2} \) |
| 13 | \( 1 + 6.46T + 13T^{2} \) |
| 17 | \( 1 + 3.38T + 17T^{2} \) |
| 19 | \( 1 - 6.62T + 19T^{2} \) |
| 23 | \( 1 - 5.53T + 23T^{2} \) |
| 29 | \( 1 + 1.01T + 29T^{2} \) |
| 31 | \( 1 + 2.72T + 31T^{2} \) |
| 37 | \( 1 + 2.15T + 37T^{2} \) |
| 41 | \( 1 - 2.81T + 41T^{2} \) |
| 43 | \( 1 - 12.4T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 8.39T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 - 5.64T + 61T^{2} \) |
| 67 | \( 1 + 3.16T + 67T^{2} \) |
| 71 | \( 1 - 4.85T + 71T^{2} \) |
| 73 | \( 1 + 0.831T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 + 4.81T + 83T^{2} \) |
| 89 | \( 1 - 8.02T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−8.681177535416154116586351189339, −7.70750262916381002348733224527, −7.33948158038544420518909465775, −6.89169831827544393796290458431, −5.37161117650915665771209262398, −4.52174169930076183380655727940, −3.60035999110064682928895223265, −2.66473927516429406839083687662, −2.21909447721546870588093376138, −1.07763915755655351865225135593,
1.07763915755655351865225135593, 2.21909447721546870588093376138, 2.66473927516429406839083687662, 3.60035999110064682928895223265, 4.52174169930076183380655727940, 5.37161117650915665771209262398, 6.89169831827544393796290458431, 7.33948158038544420518909465775, 7.70750262916381002348733224527, 8.681177535416154116586351189339