| L(s) = 1 | + 1.79·2-s − 3-s + 1.21·4-s + 1.56·5-s − 1.79·6-s − 7-s − 1.40·8-s + 9-s + 2.79·10-s + 2.49·11-s − 1.21·12-s + 0.908·13-s − 1.79·14-s − 1.56·15-s − 4.95·16-s + 1.79·18-s + 3.82·19-s + 1.90·20-s + 21-s + 4.48·22-s + 2.25·23-s + 1.40·24-s − 2.56·25-s + 1.63·26-s − 27-s − 1.21·28-s + 1.57·29-s + ⋯ |
| L(s) = 1 | + 1.26·2-s − 0.577·3-s + 0.609·4-s + 0.697·5-s − 0.732·6-s − 0.377·7-s − 0.495·8-s + 0.333·9-s + 0.885·10-s + 0.753·11-s − 0.351·12-s + 0.251·13-s − 0.479·14-s − 0.402·15-s − 1.23·16-s + 0.422·18-s + 0.877·19-s + 0.425·20-s + 0.218·21-s + 0.955·22-s + 0.470·23-s + 0.286·24-s − 0.512·25-s + 0.319·26-s − 0.192·27-s − 0.230·28-s + 0.291·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.473953360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.473953360\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 1.79T + 2T^{2} \) |
| 5 | \( 1 - 1.56T + 5T^{2} \) |
| 11 | \( 1 - 2.49T + 11T^{2} \) |
| 13 | \( 1 - 0.908T + 13T^{2} \) |
| 19 | \( 1 - 3.82T + 19T^{2} \) |
| 23 | \( 1 - 2.25T + 23T^{2} \) |
| 29 | \( 1 - 1.57T + 29T^{2} \) |
| 31 | \( 1 - 4.98T + 31T^{2} \) |
| 37 | \( 1 + 0.783T + 37T^{2} \) |
| 41 | \( 1 + 0.437T + 41T^{2} \) |
| 43 | \( 1 + 4.68T + 43T^{2} \) |
| 47 | \( 1 - 3.90T + 47T^{2} \) |
| 53 | \( 1 - 5.60T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 - 13.9T + 61T^{2} \) |
| 67 | \( 1 + 5.28T + 67T^{2} \) |
| 71 | \( 1 - 7.55T + 71T^{2} \) |
| 73 | \( 1 - 2.51T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 8.42T + 83T^{2} \) |
| 89 | \( 1 + 5.94T + 89T^{2} \) |
| 97 | \( 1 - 1.07T + 97T^{2} \) |
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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
−7.932216221087817382832113544574, −6.82810653135202140593094010565, −6.51550072673307614334198748365, −5.76162753885094498501692626049, −5.29078308585330532231447746879, −4.51878049302168311277963498908, −3.75595325915802049158846348921, −3.05158365197355961962881720916, −2.04040574249054268826538346710, −0.843622595930198859915767335810,
0.843622595930198859915767335810, 2.04040574249054268826538346710, 3.05158365197355961962881720916, 3.75595325915802049158846348921, 4.51878049302168311277963498908, 5.29078308585330532231447746879, 5.76162753885094498501692626049, 6.51550072673307614334198748365, 6.82810653135202140593094010565, 7.932216221087817382832113544574
