| L(s) = 1 | − 2-s + 2.03·3-s + 4-s + 3.48·5-s − 2.03·6-s + 1.98·7-s − 8-s + 1.13·9-s − 3.48·10-s + 1.49·11-s + 2.03·12-s + 3.26·13-s − 1.98·14-s + 7.09·15-s + 16-s + 5.08·17-s − 1.13·18-s − 3.34·19-s + 3.48·20-s + 4.03·21-s − 1.49·22-s + 23-s − 2.03·24-s + 7.17·25-s − 3.26·26-s − 3.78·27-s + 1.98·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.17·3-s + 0.5·4-s + 1.56·5-s − 0.830·6-s + 0.750·7-s − 0.353·8-s + 0.379·9-s − 1.10·10-s + 0.452·11-s + 0.587·12-s + 0.904·13-s − 0.530·14-s + 1.83·15-s + 0.250·16-s + 1.23·17-s − 0.268·18-s − 0.768·19-s + 0.780·20-s + 0.881·21-s − 0.319·22-s + 0.208·23-s − 0.415·24-s + 1.43·25-s − 0.639·26-s − 0.729·27-s + 0.375·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.822261711\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.822261711\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 - 2.03T + 3T^{2} \) |
| 5 | \( 1 - 3.48T + 5T^{2} \) |
| 7 | \( 1 - 1.98T + 7T^{2} \) |
| 11 | \( 1 - 1.49T + 11T^{2} \) |
| 13 | \( 1 - 3.26T + 13T^{2} \) |
| 17 | \( 1 - 5.08T + 17T^{2} \) |
| 19 | \( 1 + 3.34T + 19T^{2} \) |
| 29 | \( 1 - 6.06T + 29T^{2} \) |
| 31 | \( 1 - 9.51T + 31T^{2} \) |
| 37 | \( 1 + 0.908T + 37T^{2} \) |
| 41 | \( 1 - 8.64T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 + 6.22T + 47T^{2} \) |
| 53 | \( 1 + 9.86T + 53T^{2} \) |
| 59 | \( 1 - 5.32T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 - 3.94T + 79T^{2} \) |
| 83 | \( 1 + 6.93T + 83T^{2} \) |
| 89 | \( 1 + 9.93T + 89T^{2} \) |
| 97 | \( 1 - 9.14T + 97T^{2} \) |
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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
−8.249033997259260696566140744583, −7.74450998405061871236475641908, −6.53608628859969246280738157923, −6.22830614781843815660744521007, −5.32525170226723793666851543567, −4.40770811492582334922699827080, −3.22240455521665122777126522766, −2.67749608671536322996457511565, −1.68540886670635185947585331817, −1.27082790439937579409228841154,
1.27082790439937579409228841154, 1.68540886670635185947585331817, 2.67749608671536322996457511565, 3.22240455521665122777126522766, 4.40770811492582334922699827080, 5.32525170226723793666851543567, 6.22830614781843815660744521007, 6.53608628859969246280738157923, 7.74450998405061871236475641908, 8.249033997259260696566140744583
