| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 3.36·7-s − 8-s + 9-s + 4.03·11-s − 12-s + 2.91·13-s + 3.36·14-s + 16-s + 1.20·17-s − 18-s − 0.591·19-s + 3.36·21-s − 4.03·22-s − 8.53·23-s + 24-s − 2.91·26-s − 27-s − 3.36·28-s − 6.30·29-s − 0.0361·31-s − 32-s − 4.03·33-s − 1.20·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 1.27·7-s − 0.353·8-s + 0.333·9-s + 1.21·11-s − 0.288·12-s + 0.807·13-s + 0.899·14-s + 0.250·16-s + 0.293·17-s − 0.235·18-s − 0.135·19-s + 0.734·21-s − 0.860·22-s − 1.78·23-s + 0.204·24-s − 0.571·26-s − 0.192·27-s − 0.635·28-s − 1.17·29-s − 0.00649·31-s − 0.176·32-s − 0.702·33-s − 0.207·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 3.36T + 7T^{2} \) |
| 11 | \( 1 - 4.03T + 11T^{2} \) |
| 13 | \( 1 - 2.91T + 13T^{2} \) |
| 17 | \( 1 - 1.20T + 17T^{2} \) |
| 19 | \( 1 + 0.591T + 19T^{2} \) |
| 23 | \( 1 + 8.53T + 23T^{2} \) |
| 29 | \( 1 + 6.30T + 29T^{2} \) |
| 31 | \( 1 + 0.0361T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 - 2.19T + 41T^{2} \) |
| 43 | \( 1 - 9.45T + 43T^{2} \) |
| 47 | \( 1 - 5.43T + 47T^{2} \) |
| 53 | \( 1 - 6.55T + 53T^{2} \) |
| 59 | \( 1 - 0.178T + 59T^{2} \) |
| 61 | \( 1 - 0.646T + 61T^{2} \) |
| 67 | \( 1 - 9.50T + 67T^{2} \) |
| 71 | \( 1 - 8.74T + 71T^{2} \) |
| 73 | \( 1 + 16.0T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 16.4T + 83T^{2} \) |
| 89 | \( 1 + 1.40T + 89T^{2} \) |
| 97 | \( 1 + 8.63T + 97T^{2} \) |
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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
−8.200383318400314560923082427535, −7.31514377178040429033481166042, −6.60599992001677399692022379319, −6.11326745571300895563046672396, −5.48318221541314529454002954556, −3.90166622641528482648839876743, −3.70132000143096434653106430809, −2.28407058842610371251835161401, −1.19380171109081369838334441094, 0,
1.19380171109081369838334441094, 2.28407058842610371251835161401, 3.70132000143096434653106430809, 3.90166622641528482648839876743, 5.48318221541314529454002954556, 6.11326745571300895563046672396, 6.60599992001677399692022379319, 7.31514377178040429033481166042, 8.200383318400314560923082427535
