| L(s) = 1 | − 44.2·2-s + 81·3-s + 1.44e3·4-s − 3.58e3·6-s + 3.87e3·7-s − 4.14e4·8-s + 6.56e3·9-s + 8.48e4·11-s + 1.17e5·12-s + 5.86e4·13-s − 1.71e5·14-s + 1.09e6·16-s + 1.76e5·17-s − 2.90e5·18-s + 8.00e5·19-s + 3.14e5·21-s − 3.75e6·22-s − 1.65e6·23-s − 3.35e6·24-s − 2.59e6·26-s + 5.31e5·27-s + 5.61e6·28-s − 4.70e6·29-s + 5.05e6·31-s − 2.71e7·32-s + 6.86e6·33-s − 7.80e6·34-s + ⋯ |
| L(s) = 1 | − 1.95·2-s + 0.577·3-s + 2.82·4-s − 1.12·6-s + 0.610·7-s − 3.57·8-s + 0.333·9-s + 1.74·11-s + 1.63·12-s + 0.569·13-s − 1.19·14-s + 4.17·16-s + 0.511·17-s − 0.652·18-s + 1.40·19-s + 0.352·21-s − 3.41·22-s − 1.23·23-s − 2.06·24-s − 1.11·26-s + 0.192·27-s + 1.72·28-s − 1.23·29-s + 0.982·31-s − 4.58·32-s + 1.00·33-s − 1.00·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.374016191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.374016191\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 81T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 44.2T + 512T^{2} \) |
| 7 | \( 1 - 3.87e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 8.48e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 5.86e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.76e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 8.00e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.65e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.70e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.05e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 4.68e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 4.71e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.65e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.79e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.50e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.15e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 3.48e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.51e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.98e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 7.02e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.16e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 7.69e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 9.57e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 4.93e8T + 7.60e17T^{2} \) |
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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
−11.98340578208447457083151929999, −11.38433089825944468755279814029, −9.990436361482301727059952514958, −9.232209182551095918525872775547, −8.280046923849814055253957666622, −7.36621639111516527951331746945, −6.14215255004559162602801585586, −3.48452237788057898030931733347, −1.81740846936672262754247996565, −0.987694852144141500660473142921,
0.987694852144141500660473142921, 1.81740846936672262754247996565, 3.48452237788057898030931733347, 6.14215255004559162602801585586, 7.36621639111516527951331746945, 8.280046923849814055253957666622, 9.232209182551095918525872775547, 9.990436361482301727059952514958, 11.38433089825944468755279814029, 11.98340578208447457083151929999
