| L(s) = 1 | + 2-s + 0.213·3-s + 4-s − 2.70·5-s + 0.213·6-s + 1.35·7-s + 8-s − 2.95·9-s − 2.70·10-s − 0.885·11-s + 0.213·12-s − 1.69·13-s + 1.35·14-s − 0.578·15-s + 16-s + 3.88·17-s − 2.95·18-s − 7.31·19-s − 2.70·20-s + 0.290·21-s − 0.885·22-s + 1.41·23-s + 0.213·24-s + 2.32·25-s − 1.69·26-s − 1.27·27-s + 1.35·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.123·3-s + 0.5·4-s − 1.21·5-s + 0.0873·6-s + 0.513·7-s + 0.353·8-s − 0.984·9-s − 0.855·10-s − 0.266·11-s + 0.0617·12-s − 0.469·13-s + 0.363·14-s − 0.149·15-s + 0.250·16-s + 0.941·17-s − 0.696·18-s − 1.67·19-s − 0.605·20-s + 0.0634·21-s − 0.188·22-s + 0.295·23-s + 0.0436·24-s + 0.465·25-s − 0.331·26-s − 0.245·27-s + 0.256·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 - 0.213T + 3T^{2} \) |
| 5 | \( 1 + 2.70T + 5T^{2} \) |
| 7 | \( 1 - 1.35T + 7T^{2} \) |
| 11 | \( 1 + 0.885T + 11T^{2} \) |
| 13 | \( 1 + 1.69T + 13T^{2} \) |
| 17 | \( 1 - 3.88T + 17T^{2} \) |
| 19 | \( 1 + 7.31T + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 - 2.96T + 31T^{2} \) |
| 37 | \( 1 + 3.25T + 37T^{2} \) |
| 41 | \( 1 + 2.67T + 41T^{2} \) |
| 43 | \( 1 + 12.8T + 43T^{2} \) |
| 47 | \( 1 - 5.26T + 47T^{2} \) |
| 53 | \( 1 + 1.14T + 53T^{2} \) |
| 59 | \( 1 - 6.68T + 59T^{2} \) |
| 61 | \( 1 + 8.23T + 61T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 - 0.203T + 73T^{2} \) |
| 79 | \( 1 - 9.68T + 79T^{2} \) |
| 83 | \( 1 + 0.507T + 83T^{2} \) |
| 89 | \( 1 + 1.24T + 89T^{2} \) |
| 97 | \( 1 + 1.55T + 97T^{2} \) |
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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
−8.806483652685141024169041641883, −8.128018947364039341011484199429, −7.58732978219662123126039772817, −6.63197226398596061121523792816, −5.58900114492323256954343378226, −4.85343720598178998148840948654, −3.90319367878312664239065836639, −3.16773435871002878355119840291, −1.99270097484753812252728381392, 0,
1.99270097484753812252728381392, 3.16773435871002878355119840291, 3.90319367878312664239065836639, 4.85343720598178998148840948654, 5.58900114492323256954343378226, 6.63197226398596061121523792816, 7.58732978219662123126039772817, 8.128018947364039341011484199429, 8.806483652685141024169041641883
