| L(s) = 1 | + 3.30·3-s + 7.93·9-s − 5.53·11-s + 1.73·13-s − 3.73·17-s − 2.64·19-s − 4.05·23-s + 16.3·27-s − 3.81·29-s − 9.80·31-s − 18.2·33-s − 3.34·37-s + 5.75·39-s − 3.39·41-s − 10.5·43-s − 1.31·47-s − 12.3·51-s + 6.60·53-s − 8.73·57-s − 8.04·59-s − 1.03·61-s − 4.94·67-s − 13.4·69-s − 7.95·71-s − 4.39·73-s + 1.88·79-s + 30.1·81-s + ⋯ |
| L(s) = 1 | + 1.90·3-s + 2.64·9-s − 1.66·11-s + 0.482·13-s − 0.904·17-s − 0.605·19-s − 0.845·23-s + 3.14·27-s − 0.709·29-s − 1.76·31-s − 3.18·33-s − 0.549·37-s + 0.920·39-s − 0.530·41-s − 1.60·43-s − 0.191·47-s − 1.72·51-s + 0.907·53-s − 1.15·57-s − 1.04·59-s − 0.133·61-s − 0.603·67-s − 1.61·69-s − 0.943·71-s − 0.514·73-s + 0.212·79-s + 3.35·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 3.30T + 3T^{2} \) |
| 11 | \( 1 + 5.53T + 11T^{2} \) |
| 13 | \( 1 - 1.73T + 13T^{2} \) |
| 17 | \( 1 + 3.73T + 17T^{2} \) |
| 19 | \( 1 + 2.64T + 19T^{2} \) |
| 23 | \( 1 + 4.05T + 23T^{2} \) |
| 29 | \( 1 + 3.81T + 29T^{2} \) |
| 31 | \( 1 + 9.80T + 31T^{2} \) |
| 37 | \( 1 + 3.34T + 37T^{2} \) |
| 41 | \( 1 + 3.39T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 1.31T + 47T^{2} \) |
| 53 | \( 1 - 6.60T + 53T^{2} \) |
| 59 | \( 1 + 8.04T + 59T^{2} \) |
| 61 | \( 1 + 1.03T + 61T^{2} \) |
| 67 | \( 1 + 4.94T + 67T^{2} \) |
| 71 | \( 1 + 7.95T + 71T^{2} \) |
| 73 | \( 1 + 4.39T + 73T^{2} \) |
| 79 | \( 1 - 1.88T + 79T^{2} \) |
| 83 | \( 1 - 5.87T + 83T^{2} \) |
| 89 | \( 1 + 1.12T + 89T^{2} \) |
| 97 | \( 1 - 2.85T + 97T^{2} \) |
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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
−7.53179839808212616105527099467, −6.98368418856303286772054394815, −6.03871216869827234537998374660, −5.08627074055594295200580641346, −4.36765629113129115358937909927, −3.58042008947858380349435655839, −3.06320883504565844084839710841, −2.09444794816271406591330910671, −1.82687499813373476771456110908, 0,
1.82687499813373476771456110908, 2.09444794816271406591330910671, 3.06320883504565844084839710841, 3.58042008947858380349435655839, 4.36765629113129115358937909927, 5.08627074055594295200580641346, 6.03871216869827234537998374660, 6.98368418856303286772054394815, 7.53179839808212616105527099467
