| L(s) = 1 | + 1.88·2-s − 0.403·3-s + 1.53·4-s − 2.96·5-s − 0.757·6-s + 2.73·7-s − 0.875·8-s − 2.83·9-s − 5.56·10-s + 11-s − 0.618·12-s − 4.24·13-s + 5.14·14-s + 1.19·15-s − 4.71·16-s − 17-s − 5.33·18-s + 4.36·19-s − 4.54·20-s − 1.10·21-s + 1.88·22-s − 6.82·23-s + 0.352·24-s + 3.77·25-s − 7.98·26-s + 2.35·27-s + 4.20·28-s + ⋯ |
| L(s) = 1 | + 1.32·2-s − 0.232·3-s + 0.767·4-s − 1.32·5-s − 0.309·6-s + 1.03·7-s − 0.309·8-s − 0.945·9-s − 1.76·10-s + 0.301·11-s − 0.178·12-s − 1.17·13-s + 1.37·14-s + 0.308·15-s − 1.17·16-s − 0.242·17-s − 1.25·18-s + 1.00·19-s − 1.01·20-s − 0.240·21-s + 0.400·22-s − 1.42·23-s + 0.0720·24-s + 0.755·25-s − 1.56·26-s + 0.452·27-s + 0.793·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.960967111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.960967111\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 - 1.88T + 2T^{2} \) |
| 3 | \( 1 + 0.403T + 3T^{2} \) |
| 5 | \( 1 + 2.96T + 5T^{2} \) |
| 7 | \( 1 - 2.73T + 7T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 19 | \( 1 - 4.36T + 19T^{2} \) |
| 23 | \( 1 + 6.82T + 23T^{2} \) |
| 29 | \( 1 + 0.653T + 29T^{2} \) |
| 31 | \( 1 - 8.20T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 - 6.51T + 41T^{2} \) |
| 47 | \( 1 - 8.54T + 47T^{2} \) |
| 53 | \( 1 - 2.99T + 53T^{2} \) |
| 59 | \( 1 + 2.20T + 59T^{2} \) |
| 61 | \( 1 - 2.30T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 - 8.74T + 73T^{2} \) |
| 79 | \( 1 + 15.6T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 + 6.78T + 89T^{2} \) |
| 97 | \( 1 - 4.62T + 97T^{2} \) |
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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
−7.80582144042150994513553584985, −7.05338138355497239221615576189, −6.30350766489578743514742139003, −5.35704399305788043786619225889, −5.07675438010397069997145026839, −4.26399534251641197965869462708, −3.79344898057758427012088259741, −2.91624546234628418406524806677, −2.14317391495717473902616994087, −0.54550115850670026456804745866,
0.54550115850670026456804745866, 2.14317391495717473902616994087, 2.91624546234628418406524806677, 3.79344898057758427012088259741, 4.26399534251641197965869462708, 5.07675438010397069997145026839, 5.35704399305788043786619225889, 6.30350766489578743514742139003, 7.05338138355497239221615576189, 7.80582144042150994513553584985
