| L(s) = 1 | − 2.26·3-s + 2.03·5-s − 4.86·7-s + 2.14·9-s − 0.629·11-s + 0.799·13-s − 4.60·15-s − 4.64·17-s + 11.0·21-s + 8.96·23-s − 0.869·25-s + 1.94·27-s + 6.31·29-s − 9.70·31-s + 1.42·33-s − 9.88·35-s + 8.43·37-s − 1.81·39-s + 5.66·41-s + 3.80·43-s + 4.35·45-s − 4.01·47-s + 16.6·49-s + 10.5·51-s − 1.25·53-s − 1.27·55-s + 10.2·59-s + ⋯ |
| L(s) = 1 | − 1.30·3-s + 0.908·5-s − 1.83·7-s + 0.714·9-s − 0.189·11-s + 0.221·13-s − 1.19·15-s − 1.12·17-s + 2.40·21-s + 1.86·23-s − 0.173·25-s + 0.373·27-s + 1.17·29-s − 1.74·31-s + 0.248·33-s − 1.67·35-s + 1.38·37-s − 0.290·39-s + 0.884·41-s + 0.579·43-s + 0.649·45-s − 0.585·47-s + 2.38·49-s + 1.47·51-s − 0.172·53-s − 0.172·55-s + 1.33·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2.26T + 3T^{2} \) |
| 5 | \( 1 - 2.03T + 5T^{2} \) |
| 7 | \( 1 + 4.86T + 7T^{2} \) |
| 11 | \( 1 + 0.629T + 11T^{2} \) |
| 13 | \( 1 - 0.799T + 13T^{2} \) |
| 17 | \( 1 + 4.64T + 17T^{2} \) |
| 23 | \( 1 - 8.96T + 23T^{2} \) |
| 29 | \( 1 - 6.31T + 29T^{2} \) |
| 31 | \( 1 + 9.70T + 31T^{2} \) |
| 37 | \( 1 - 8.43T + 37T^{2} \) |
| 41 | \( 1 - 5.66T + 41T^{2} \) |
| 43 | \( 1 - 3.80T + 43T^{2} \) |
| 47 | \( 1 + 4.01T + 47T^{2} \) |
| 53 | \( 1 + 1.25T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 + 1.26T + 67T^{2} \) |
| 71 | \( 1 - 8.22T + 71T^{2} \) |
| 73 | \( 1 + 6.77T + 73T^{2} \) |
| 79 | \( 1 + 6.88T + 79T^{2} \) |
| 83 | \( 1 - 0.202T + 83T^{2} \) |
| 89 | \( 1 + 7.54T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−7.38506139993273520753755054910, −6.75570004048966741950307470796, −6.23856897601659832619993836573, −5.82565698382268848367081103552, −5.08044070771442529921884655143, −4.21743284420118388271393310170, −3.12957862936032662162623650784, −2.42816956985931699059485765201, −1.01745574867565953364286325701, 0,
1.01745574867565953364286325701, 2.42816956985931699059485765201, 3.12957862936032662162623650784, 4.21743284420118388271393310170, 5.08044070771442529921884655143, 5.82565698382268848367081103552, 6.23856897601659832619993836573, 6.75570004048966741950307470796, 7.38506139993273520753755054910
