| L(s) = 1 | + 2.39·2-s + 3.74·4-s − 5-s − 2.89·7-s + 4.19·8-s − 2.39·10-s + 11-s + 4.73·13-s − 6.93·14-s + 2.56·16-s + 5.65·17-s + 19-s − 3.74·20-s + 2.39·22-s + 4.00·23-s + 25-s + 11.3·26-s − 10.8·28-s − 9.32·29-s − 6.60·31-s − 2.24·32-s + 13.5·34-s + 2.89·35-s + 6.07·37-s + 2.39·38-s − 4.19·40-s − 5.47·41-s + ⋯ |
| L(s) = 1 | + 1.69·2-s + 1.87·4-s − 0.447·5-s − 1.09·7-s + 1.48·8-s − 0.758·10-s + 0.301·11-s + 1.31·13-s − 1.85·14-s + 0.640·16-s + 1.37·17-s + 0.229·19-s − 0.838·20-s + 0.511·22-s + 0.835·23-s + 0.200·25-s + 2.22·26-s − 2.05·28-s − 1.73·29-s − 1.18·31-s − 0.397·32-s + 2.32·34-s + 0.489·35-s + 0.999·37-s + 0.388·38-s − 0.663·40-s − 0.854·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.447718874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.447718874\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.39T + 2T^{2} \) |
| 7 | \( 1 + 2.89T + 7T^{2} \) |
| 13 | \( 1 - 4.73T + 13T^{2} \) |
| 17 | \( 1 - 5.65T + 17T^{2} \) |
| 23 | \( 1 - 4.00T + 23T^{2} \) |
| 29 | \( 1 + 9.32T + 29T^{2} \) |
| 31 | \( 1 + 6.60T + 31T^{2} \) |
| 37 | \( 1 - 6.07T + 37T^{2} \) |
| 41 | \( 1 + 5.47T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 - 0.295T + 47T^{2} \) |
| 53 | \( 1 - 3.81T + 53T^{2} \) |
| 59 | \( 1 - 5.54T + 59T^{2} \) |
| 61 | \( 1 + 1.01T + 61T^{2} \) |
| 67 | \( 1 - 6.98T + 67T^{2} \) |
| 71 | \( 1 - 1.02T + 71T^{2} \) |
| 73 | \( 1 + 0.202T + 73T^{2} \) |
| 79 | \( 1 - 7.28T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 - 4.81T + 97T^{2} \) |
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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
−7.33908764996712674220205317201, −6.87204956272729777408502061590, −5.99287652759232128939669207394, −5.74988290362478611505772049188, −4.96128856330064712339818867161, −3.92940805541184765348540977539, −3.57190785743125325960999159437, −3.16317972728223342266884101245, −2.07283915293441647267145009974, −0.869743060948438292399171644980,
0.869743060948438292399171644980, 2.07283915293441647267145009974, 3.16317972728223342266884101245, 3.57190785743125325960999159437, 3.92940805541184765348540977539, 4.96128856330064712339818867161, 5.74988290362478611505772049188, 5.99287652759232128939669207394, 6.87204956272729777408502061590, 7.33908764996712674220205317201
