| L(s) = 1 | + 1.44·3-s − 0.898·9-s − 0.550·11-s + 7.89·17-s − 8.34·19-s − 5.65·27-s − 0.797·33-s − 12.7·41-s − 10·43-s − 7·49-s + 11.4·51-s − 12.1·57-s + 6·59-s + 14.3·67-s − 13.6·73-s − 5.49·81-s − 11.4·83-s + 13.8·89-s + 10·97-s + 0.494·99-s − 20.1·107-s + 0.797·113-s + ⋯ |
| L(s) = 1 | + 0.836·3-s − 0.299·9-s − 0.165·11-s + 1.91·17-s − 1.91·19-s − 1.08·27-s − 0.138·33-s − 1.99·41-s − 1.52·43-s − 49-s + 1.60·51-s − 1.60·57-s + 0.781·59-s + 1.75·67-s − 1.60·73-s − 0.610·81-s − 1.25·83-s + 1.47·89-s + 1.01·97-s + 0.0497·99-s − 1.94·107-s + 0.0750·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 - 1.44T + 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 0.550T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 7.89T + 17T^{2} \) |
| 19 | \( 1 + 8.34T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 12.7T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
| 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
−7.939724002771569385501129680407, −7.02913034123628046649906047335, −6.30728773965330542984906644477, −5.51439306110800516733455869480, −4.81256220073585316426354063260, −3.73354091258120423012028578223, −3.26941960389057798465229049737, −2.36364980743719547641105914999, −1.51981137106982608364168941711, 0,
1.51981137106982608364168941711, 2.36364980743719547641105914999, 3.26941960389057798465229049737, 3.73354091258120423012028578223, 4.81256220073585316426354063260, 5.51439306110800516733455869480, 6.30728773965330542984906644477, 7.02913034123628046649906047335, 7.939724002771569385501129680407
