| L(s) = 1 | − 1.49·2-s + 2.11·3-s + 0.242·4-s − 5-s − 3.16·6-s + 4.23·7-s + 2.63·8-s + 1.47·9-s + 1.49·10-s + 11-s + 0.512·12-s + 0.374·13-s − 6.34·14-s − 2.11·15-s − 4.42·16-s − 0.922·17-s − 2.20·18-s − 1.09·19-s − 0.242·20-s + 8.96·21-s − 1.49·22-s + 7.06·23-s + 5.56·24-s + 25-s − 0.560·26-s − 3.22·27-s + 1.02·28-s + ⋯ |
| L(s) = 1 | − 1.05·2-s + 1.22·3-s + 0.121·4-s − 0.447·5-s − 1.29·6-s + 1.60·7-s + 0.930·8-s + 0.491·9-s + 0.473·10-s + 0.301·11-s + 0.147·12-s + 0.103·13-s − 1.69·14-s − 0.546·15-s − 1.10·16-s − 0.223·17-s − 0.520·18-s − 0.251·19-s − 0.0541·20-s + 1.95·21-s − 0.319·22-s + 1.47·23-s + 1.13·24-s + 0.200·25-s − 0.110·26-s − 0.621·27-s + 0.193·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.861602350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.861602350\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 + 1.49T + 2T^{2} \) |
| 3 | \( 1 - 2.11T + 3T^{2} \) |
| 7 | \( 1 - 4.23T + 7T^{2} \) |
| 13 | \( 1 - 0.374T + 13T^{2} \) |
| 17 | \( 1 + 0.922T + 17T^{2} \) |
| 19 | \( 1 + 1.09T + 19T^{2} \) |
| 23 | \( 1 - 7.06T + 23T^{2} \) |
| 29 | \( 1 - 1.65T + 29T^{2} \) |
| 31 | \( 1 - 3.32T + 31T^{2} \) |
| 37 | \( 1 - 1.95T + 37T^{2} \) |
| 41 | \( 1 - 3.76T + 41T^{2} \) |
| 43 | \( 1 + 5.75T + 43T^{2} \) |
| 47 | \( 1 - 3.00T + 47T^{2} \) |
| 53 | \( 1 - 4.60T + 53T^{2} \) |
| 59 | \( 1 + 7.36T + 59T^{2} \) |
| 61 | \( 1 + 3.22T + 61T^{2} \) |
| 67 | \( 1 - 6.43T + 67T^{2} \) |
| 71 | \( 1 - 9.26T + 71T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 - 2.91T + 83T^{2} \) |
| 89 | \( 1 - 0.554T + 89T^{2} \) |
| 97 | \( 1 + 5.85T + 97T^{2} \) |
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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
−8.444986621173727245059099470032, −7.985822248684138984630392513380, −7.49193626508249803041430934524, −6.66182727362413712100099304727, −5.22595926823638097203182597579, −4.56479948012761794749598618984, −3.83809530446026639092333422149, −2.70596696569632348428193627445, −1.80949603649744345141796857114, −0.929623265994020573001180967016,
0.929623265994020573001180967016, 1.80949603649744345141796857114, 2.70596696569632348428193627445, 3.83809530446026639092333422149, 4.56479948012761794749598618984, 5.22595926823638097203182597579, 6.66182727362413712100099304727, 7.49193626508249803041430934524, 7.985822248684138984630392513380, 8.444986621173727245059099470032
