| L(s) = 1 | − 3-s + 2.97·7-s + 9-s + 5.59·11-s + 6.43·13-s − 1.61·17-s − 3.07·19-s − 2.97·21-s + 3.59·23-s − 27-s + 7.19·29-s + 5.69·31-s − 5.59·33-s − 9.27·37-s − 6.43·39-s + 11.4·41-s − 0.145·43-s − 12.9·47-s + 1.86·49-s + 1.61·51-s + 0.419·53-s + 3.07·57-s + 5.54·59-s − 11.8·61-s + 2.97·63-s − 6.91·67-s − 3.59·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.12·7-s + 0.333·9-s + 1.68·11-s + 1.78·13-s − 0.392·17-s − 0.705·19-s − 0.649·21-s + 0.749·23-s − 0.192·27-s + 1.33·29-s + 1.02·31-s − 0.973·33-s − 1.52·37-s − 1.03·39-s + 1.78·41-s − 0.0222·43-s − 1.88·47-s + 0.266·49-s + 0.226·51-s + 0.0575·53-s + 0.407·57-s + 0.721·59-s − 1.51·61-s + 0.375·63-s − 0.844·67-s − 0.432·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.224129816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.224129816\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2.97T + 7T^{2} \) |
| 11 | \( 1 - 5.59T + 11T^{2} \) |
| 13 | \( 1 - 6.43T + 13T^{2} \) |
| 17 | \( 1 + 1.61T + 17T^{2} \) |
| 19 | \( 1 + 3.07T + 19T^{2} \) |
| 23 | \( 1 - 3.59T + 23T^{2} \) |
| 29 | \( 1 - 7.19T + 29T^{2} \) |
| 31 | \( 1 - 5.69T + 31T^{2} \) |
| 37 | \( 1 + 9.27T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 + 0.145T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 - 0.419T + 53T^{2} \) |
| 59 | \( 1 - 5.54T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 + 6.91T + 67T^{2} \) |
| 71 | \( 1 + 6.81T + 71T^{2} \) |
| 73 | \( 1 + 3.33T + 73T^{2} \) |
| 79 | \( 1 + 6.30T + 79T^{2} \) |
| 83 | \( 1 + 8.45T + 83T^{2} \) |
| 89 | \( 1 + 2.31T + 89T^{2} \) |
| 97 | \( 1 - 9.44T + 97T^{2} \) |
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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
−8.640582582411794763059685300342, −8.209188911578849921976864446441, −7.00749505004139334689977611590, −6.41730067896646029000021502887, −5.84870253779949914775045033368, −4.64825458446187361711626019665, −4.27384491574508148582746841115, −3.22536901112793357792905790580, −1.66730771588660234593449350880, −1.08563774430134004284320098974,
1.08563774430134004284320098974, 1.66730771588660234593449350880, 3.22536901112793357792905790580, 4.27384491574508148582746841115, 4.64825458446187361711626019665, 5.84870253779949914775045033368, 6.41730067896646029000021502887, 7.00749505004139334689977611590, 8.209188911578849921976864446441, 8.640582582411794763059685300342
