| L(s) = 1 | + 9.66e16·2-s + 4.03e28·3-s − 6.55e35·4-s − 8.26e40·5-s + 3.89e45·6-s − 7.51e49·7-s − 1.27e53·8-s + 1.02e57·9-s − 7.98e57·10-s + 1.61e62·11-s − 2.64e64·12-s + 4.29e65·13-s − 7.26e66·14-s − 3.32e69·15-s + 4.23e71·16-s − 1.13e73·17-s + 9.90e73·18-s − 6.93e75·19-s + 5.41e76·20-s − 3.02e78·21-s + 1.55e79·22-s − 1.33e81·23-s − 5.14e81·24-s − 1.43e83·25-s + 4.14e82·26-s + 1.71e85·27-s + 4.92e85·28-s + ⋯ |
| L(s) = 1 | + 0.118·2-s + 1.64·3-s − 0.985·4-s − 0.212·5-s + 0.195·6-s − 0.391·7-s − 0.235·8-s + 1.71·9-s − 0.0252·10-s + 1.75·11-s − 1.62·12-s + 0.225·13-s − 0.0463·14-s − 0.350·15-s + 0.958·16-s − 0.696·17-s + 0.202·18-s − 0.569·19-s + 0.209·20-s − 0.644·21-s + 0.208·22-s − 1.26·23-s − 0.387·24-s − 0.954·25-s + 0.0267·26-s + 1.17·27-s + 0.385·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(120-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+59.5) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(60)\) |
\(\approx\) |
\(3.458955660\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.458955660\) |
| \(L(\frac{121}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 9.66e16T + 6.64e35T^{2} \) |
| 3 | \( 1 - 4.03e28T + 5.99e56T^{2} \) |
| 5 | \( 1 + 8.26e40T + 1.50e83T^{2} \) |
| 7 | \( 1 + 7.51e49T + 3.68e100T^{2} \) |
| 11 | \( 1 - 1.61e62T + 8.42e123T^{2} \) |
| 13 | \( 1 - 4.29e65T + 3.62e132T^{2} \) |
| 17 | \( 1 + 1.13e73T + 2.65e146T^{2} \) |
| 19 | \( 1 + 6.93e75T + 1.48e152T^{2} \) |
| 23 | \( 1 + 1.33e81T + 1.11e162T^{2} \) |
| 29 | \( 1 - 1.18e87T + 1.06e174T^{2} \) |
| 31 | \( 1 - 9.34e88T + 2.96e177T^{2} \) |
| 37 | \( 1 + 1.79e93T + 4.13e186T^{2} \) |
| 41 | \( 1 - 1.07e96T + 8.34e191T^{2} \) |
| 43 | \( 1 - 1.25e97T + 2.41e194T^{2} \) |
| 47 | \( 1 - 3.63e99T + 9.54e198T^{2} \) |
| 53 | \( 1 - 5.76e102T + 1.54e205T^{2} \) |
| 59 | \( 1 + 3.83e104T + 5.38e210T^{2} \) |
| 61 | \( 1 - 1.59e106T + 2.84e212T^{2} \) |
| 67 | \( 1 - 4.96e108T + 2.00e217T^{2} \) |
| 71 | \( 1 - 4.39e109T + 1.99e220T^{2} \) |
| 73 | \( 1 - 1.46e111T + 5.43e221T^{2} \) |
| 79 | \( 1 - 1.06e113T + 6.57e225T^{2} \) |
| 83 | \( 1 + 1.10e114T + 2.34e228T^{2} \) |
| 89 | \( 1 - 6.21e115T + 9.49e231T^{2} \) |
| 97 | \( 1 - 7.16e117T + 2.66e236T^{2} \) |
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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
−13.69237084500605849519667045502, −12.23305264718153101420004976248, −9.854200467251363106310856188359, −8.954011733008356998308746988973, −8.133621242561840574080051291465, −6.44340444128789625083748795899, −4.19620828612214647521515968142, −3.75304219474267633330360321094, −2.30072654806638291515999167999, −0.868840083626963404525161336414,
0.868840083626963404525161336414, 2.30072654806638291515999167999, 3.75304219474267633330360321094, 4.19620828612214647521515968142, 6.44340444128789625083748795899, 8.133621242561840574080051291465, 8.954011733008356998308746988973, 9.854200467251363106310856188359, 12.23305264718153101420004976248, 13.69237084500605849519667045502
