| L(s) = 1 | − 1.90·2-s + 3-s + 1.61·4-s − 5-s − 1.90·6-s − 1.51·7-s + 0.724·8-s + 9-s + 1.90·10-s + 6.31·11-s + 1.61·12-s + 2.69·13-s + 2.87·14-s − 15-s − 4.61·16-s − 5.81·17-s − 1.90·18-s + 6.72·19-s − 1.61·20-s − 1.51·21-s − 12.0·22-s + 1.50·23-s + 0.724·24-s + 25-s − 5.13·26-s + 27-s − 2.45·28-s + ⋯ |
| L(s) = 1 | − 1.34·2-s + 0.577·3-s + 0.809·4-s − 0.447·5-s − 0.776·6-s − 0.572·7-s + 0.256·8-s + 0.333·9-s + 0.601·10-s + 1.90·11-s + 0.467·12-s + 0.748·13-s + 0.769·14-s − 0.258·15-s − 1.15·16-s − 1.40·17-s − 0.448·18-s + 1.54·19-s − 0.362·20-s − 0.330·21-s − 2.56·22-s + 0.313·23-s + 0.147·24-s + 0.200·25-s − 1.00·26-s + 0.192·27-s − 0.463·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.082644690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.082644690\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 163 | \( 1 + T \) |
| good | 2 | \( 1 + 1.90T + 2T^{2} \) |
| 7 | \( 1 + 1.51T + 7T^{2} \) |
| 11 | \( 1 - 6.31T + 11T^{2} \) |
| 13 | \( 1 - 2.69T + 13T^{2} \) |
| 17 | \( 1 + 5.81T + 17T^{2} \) |
| 19 | \( 1 - 6.72T + 19T^{2} \) |
| 23 | \( 1 - 1.50T + 23T^{2} \) |
| 29 | \( 1 - 6.33T + 29T^{2} \) |
| 31 | \( 1 - 0.413T + 31T^{2} \) |
| 37 | \( 1 + 4.62T + 37T^{2} \) |
| 41 | \( 1 - 2.40T + 41T^{2} \) |
| 43 | \( 1 + 7.79T + 43T^{2} \) |
| 47 | \( 1 + 2.55T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 + 7.79T + 59T^{2} \) |
| 61 | \( 1 + 8.05T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 + 3.94T + 71T^{2} \) |
| 73 | \( 1 + 6.99T + 73T^{2} \) |
| 79 | \( 1 - 2.23T + 79T^{2} \) |
| 83 | \( 1 - 15.9T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−8.963157752887622507490406034025, −8.484912650069246779229869324219, −7.58392764018148134009454042218, −6.76831412071400379883557111203, −6.42577852316474018767215596224, −4.80146038549672274044909417040, −3.92201891831025320391417304895, −3.13281921281982238746464721315, −1.74550958858557506665697470452, −0.843453034963790395386619503406,
0.843453034963790395386619503406, 1.74550958858557506665697470452, 3.13281921281982238746464721315, 3.92201891831025320391417304895, 4.80146038549672274044909417040, 6.42577852316474018767215596224, 6.76831412071400379883557111203, 7.58392764018148134009454042218, 8.484912650069246779229869324219, 8.963157752887622507490406034025
