| L(s) = 1 | + 3.26·3-s + 2.75·5-s − 7-s + 7.68·9-s − 11-s + 13-s + 8.99·15-s + 6.66·17-s + 2.23·19-s − 3.26·21-s + 5.35·23-s + 2.57·25-s + 15.2·27-s + 2.68·29-s − 6.52·31-s − 3.26·33-s − 2.75·35-s − 5.88·37-s + 3.26·39-s − 9.57·41-s − 7.83·43-s + 21.1·45-s + 12.5·47-s + 49-s + 21.7·51-s − 3.27·53-s − 2.75·55-s + ⋯ |
| L(s) = 1 | + 1.88·3-s + 1.23·5-s − 0.377·7-s + 2.56·9-s − 0.301·11-s + 0.277·13-s + 2.32·15-s + 1.61·17-s + 0.512·19-s − 0.713·21-s + 1.11·23-s + 0.515·25-s + 2.94·27-s + 0.498·29-s − 1.17·31-s − 0.568·33-s − 0.465·35-s − 0.967·37-s + 0.523·39-s − 1.49·41-s − 1.19·43-s + 3.15·45-s + 1.83·47-s + 0.142·49-s + 3.04·51-s − 0.450·53-s − 0.371·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.035592835\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.035592835\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 3.26T + 3T^{2} \) |
| 5 | \( 1 - 2.75T + 5T^{2} \) |
| 17 | \( 1 - 6.66T + 17T^{2} \) |
| 19 | \( 1 - 2.23T + 19T^{2} \) |
| 23 | \( 1 - 5.35T + 23T^{2} \) |
| 29 | \( 1 - 2.68T + 29T^{2} \) |
| 31 | \( 1 + 6.52T + 31T^{2} \) |
| 37 | \( 1 + 5.88T + 37T^{2} \) |
| 41 | \( 1 + 9.57T + 41T^{2} \) |
| 43 | \( 1 + 7.83T + 43T^{2} \) |
| 47 | \( 1 - 12.5T + 47T^{2} \) |
| 53 | \( 1 + 3.27T + 53T^{2} \) |
| 59 | \( 1 + 7.10T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 - 2.43T + 89T^{2} \) |
| 97 | \( 1 + 5.98T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.895473334031523522105707801491, −7.23479970067117951634545499817, −6.71970205027004160657087404395, −5.62476448809619821116017883670, −5.13721555569482861456820076265, −3.97384653031142812686424228126, −3.20204844986038276455156085867, −2.85865572100071437125409908532, −1.84463329125772127660131712045, −1.26738525984837892210327214457,
1.26738525984837892210327214457, 1.84463329125772127660131712045, 2.85865572100071437125409908532, 3.20204844986038276455156085867, 3.97384653031142812686424228126, 5.13721555569482861456820076265, 5.62476448809619821116017883670, 6.71970205027004160657087404395, 7.23479970067117951634545499817, 7.895473334031523522105707801491
