L(s) = 1 | − 0.237·3-s + 2.74·5-s − 2.88·7-s − 2.94·9-s − 2.31·11-s + 0.899·13-s − 0.651·15-s + 17-s + 8.66·19-s + 0.682·21-s + 2.21·23-s + 2.55·25-s + 1.40·27-s − 4.31·29-s − 1.36·31-s + 0.548·33-s − 7.91·35-s − 4.03·37-s − 0.213·39-s − 8.30·41-s − 0.582·43-s − 8.09·45-s + 3.69·47-s + 1.29·49-s − 0.237·51-s + 1.86·53-s − 6.35·55-s + ⋯ |
L(s) = 1 | − 0.136·3-s + 1.22·5-s − 1.08·7-s − 0.981·9-s − 0.697·11-s + 0.249·13-s − 0.168·15-s + 0.242·17-s + 1.98·19-s + 0.149·21-s + 0.461·23-s + 0.511·25-s + 0.271·27-s − 0.801·29-s − 0.244·31-s + 0.0954·33-s − 1.33·35-s − 0.663·37-s − 0.0341·39-s − 1.29·41-s − 0.0888·43-s − 1.20·45-s + 0.539·47-s + 0.185·49-s − 0.0331·51-s + 0.255·53-s − 0.857·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 + 0.237T + 3T^{2} \) |
| 5 | \( 1 - 2.74T + 5T^{2} \) |
| 7 | \( 1 + 2.88T + 7T^{2} \) |
| 11 | \( 1 + 2.31T + 11T^{2} \) |
| 13 | \( 1 - 0.899T + 13T^{2} \) |
| 19 | \( 1 - 8.66T + 19T^{2} \) |
| 23 | \( 1 - 2.21T + 23T^{2} \) |
| 29 | \( 1 + 4.31T + 29T^{2} \) |
| 31 | \( 1 + 1.36T + 31T^{2} \) |
| 37 | \( 1 + 4.03T + 37T^{2} \) |
| 41 | \( 1 + 8.30T + 41T^{2} \) |
| 43 | \( 1 + 0.582T + 43T^{2} \) |
| 47 | \( 1 - 3.69T + 47T^{2} \) |
| 53 | \( 1 - 1.86T + 53T^{2} \) |
| 61 | \( 1 + 5.61T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 - 0.209T + 71T^{2} \) |
| 73 | \( 1 - 0.762T + 73T^{2} \) |
| 79 | \( 1 - 3.50T + 79T^{2} \) |
| 83 | \( 1 + 7.71T + 83T^{2} \) |
| 89 | \( 1 + 3.55T + 89T^{2} \) |
| 97 | \( 1 + 18.9T + 97T^{2} \) |
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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
−7.36354882348282699419028229327, −6.71581920889384276481679457649, −5.95438346991251131613243342459, −5.46430278676044442196058342520, −5.08472031507753647935940525033, −3.58504193068929037443210582974, −3.08996714296318743837707609564, −2.35585734471808887710919479729, −1.26466718560735549068699678280, 0,
1.26466718560735549068699678280, 2.35585734471808887710919479729, 3.08996714296318743837707609564, 3.58504193068929037443210582974, 5.08472031507753647935940525033, 5.46430278676044442196058342520, 5.95438346991251131613243342459, 6.71581920889384276481679457649, 7.36354882348282699419028229327