| L(s) = 1 | + 2.75e18·2-s + 3.25e29·3-s − 3.02e36·4-s + 1.58e43·5-s + 8.98e47·6-s + 1.53e52·7-s − 3.76e55·8-s + 5.75e58·9-s + 4.37e61·10-s + 8.10e63·11-s − 9.83e65·12-s − 2.32e68·13-s + 4.22e70·14-s + 5.16e72·15-s − 7.18e73·16-s + 5.03e75·17-s + 1.58e77·18-s + 4.63e78·19-s − 4.79e79·20-s + 4.98e81·21-s + 2.23e82·22-s − 8.75e83·23-s − 1.22e85·24-s + 1.57e86·25-s − 6.40e86·26-s + 2.92e87·27-s − 4.62e88·28-s + ⋯ |
| L(s) = 1 | + 0.846·2-s + 1.47·3-s − 0.284·4-s + 1.63·5-s + 1.25·6-s + 1.62·7-s − 1.08·8-s + 1.18·9-s + 1.38·10-s + 0.729·11-s − 0.420·12-s − 0.722·13-s + 1.37·14-s + 2.41·15-s − 0.635·16-s + 1.07·17-s + 1.00·18-s + 1.05·19-s − 0.464·20-s + 2.40·21-s + 0.617·22-s − 1.57·23-s − 1.60·24-s + 1.67·25-s − 0.610·26-s + 0.273·27-s − 0.462·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(124-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+61.5) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(62)\) |
\(\approx\) |
\(8.645625021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.645625021\) |
| \(L(\frac{125}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 2.75e18T + 1.06e37T^{2} \) |
| 3 | \( 1 - 3.25e29T + 4.85e58T^{2} \) |
| 5 | \( 1 - 1.58e43T + 9.40e85T^{2} \) |
| 7 | \( 1 - 1.53e52T + 8.85e103T^{2} \) |
| 11 | \( 1 - 8.10e63T + 1.23e128T^{2} \) |
| 13 | \( 1 + 2.32e68T + 1.03e137T^{2} \) |
| 17 | \( 1 - 5.03e75T + 2.21e151T^{2} \) |
| 19 | \( 1 - 4.63e78T + 1.93e157T^{2} \) |
| 23 | \( 1 + 8.75e83T + 3.10e167T^{2} \) |
| 29 | \( 1 + 2.83e89T + 7.49e179T^{2} \) |
| 31 | \( 1 + 9.43e90T + 2.73e183T^{2} \) |
| 37 | \( 1 + 2.53e96T + 7.74e192T^{2} \) |
| 41 | \( 1 + 7.60e97T + 2.35e198T^{2} \) |
| 43 | \( 1 + 3.52e99T + 8.25e200T^{2} \) |
| 47 | \( 1 - 1.81e102T + 4.65e205T^{2} \) |
| 53 | \( 1 + 6.68e105T + 1.21e212T^{2} \) |
| 59 | \( 1 + 9.20e108T + 6.52e217T^{2} \) |
| 61 | \( 1 + 7.18e109T + 3.94e219T^{2} \) |
| 67 | \( 1 - 3.47e111T + 4.04e224T^{2} \) |
| 71 | \( 1 - 2.22e113T + 5.06e227T^{2} \) |
| 73 | \( 1 + 2.81e114T + 1.54e229T^{2} \) |
| 79 | \( 1 - 1.01e116T + 2.55e233T^{2} \) |
| 83 | \( 1 - 1.38e118T + 1.11e236T^{2} \) |
| 89 | \( 1 - 5.06e119T + 5.95e239T^{2} \) |
| 97 | \( 1 - 8.04e121T + 2.36e244T^{2} \) |
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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
−13.80727175464027541267467493772, −12.13040406944727993676873014158, −9.874102557447802404158469160878, −9.013308724210557924536571098326, −7.76863763896603411097673964341, −5.77609261314513175012153505592, −4.77655370369272812513703915901, −3.42173622080010934887505999387, −2.18758970086827677848956539183, −1.43178416755699381070385352359,
1.43178416755699381070385352359, 2.18758970086827677848956539183, 3.42173622080010934887505999387, 4.77655370369272812513703915901, 5.77609261314513175012153505592, 7.76863763896603411097673964341, 9.013308724210557924536571098326, 9.874102557447802404158469160878, 12.13040406944727993676873014158, 13.80727175464027541267467493772
