| L(s) = 1 | + 1.39·3-s + 4.34·5-s − 1.84·7-s − 1.05·9-s − 2.67·11-s − 2.77·13-s + 6.06·15-s − 17-s − 7.09·19-s − 2.57·21-s + 6.44·23-s + 13.9·25-s − 5.65·27-s − 2.75·29-s − 0.324·31-s − 3.72·33-s − 8.01·35-s − 9.98·37-s − 3.86·39-s − 8.93·41-s + 5.64·43-s − 4.58·45-s + 7.13·47-s − 3.60·49-s − 1.39·51-s − 9.95·53-s − 11.6·55-s + ⋯ |
| L(s) = 1 | + 0.805·3-s + 1.94·5-s − 0.696·7-s − 0.351·9-s − 0.805·11-s − 0.769·13-s + 1.56·15-s − 0.242·17-s − 1.62·19-s − 0.560·21-s + 1.34·23-s + 2.78·25-s − 1.08·27-s − 0.511·29-s − 0.0582·31-s − 0.648·33-s − 1.35·35-s − 1.64·37-s − 0.619·39-s − 1.39·41-s + 0.860·43-s − 0.683·45-s + 1.04·47-s − 0.514·49-s − 0.195·51-s − 1.36·53-s − 1.56·55-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 | \( 1 - 1.39T + 3T^{2} \) |
| 5 | \( 1 - 4.34T + 5T^{2} \) |
| 7 | \( 1 + 1.84T + 7T^{2} \) |
| 11 | \( 1 + 2.67T + 11T^{2} \) |
| 13 | \( 1 + 2.77T + 13T^{2} \) |
| 19 | \( 1 + 7.09T + 19T^{2} \) |
| 23 | \( 1 - 6.44T + 23T^{2} \) |
| 29 | \( 1 + 2.75T + 29T^{2} \) |
| 31 | \( 1 + 0.324T + 31T^{2} \) |
| 37 | \( 1 + 9.98T + 37T^{2} \) |
| 41 | \( 1 + 8.93T + 41T^{2} \) |
| 43 | \( 1 - 5.64T + 43T^{2} \) |
| 47 | \( 1 - 7.13T + 47T^{2} \) |
| 53 | \( 1 + 9.95T + 53T^{2} \) |
| 61 | \( 1 - 0.667T + 61T^{2} \) |
| 67 | \( 1 - 6.42T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 + 2.90T + 73T^{2} \) |
| 79 | \( 1 + 9.78T + 79T^{2} \) |
| 83 | \( 1 - 8.62T + 83T^{2} \) |
| 89 | \( 1 - 0.128T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−7.40931959821777790566999050955, −6.67426052034421426303493044464, −6.20380229374188543521779153928, −5.30143248916480353046201263559, −4.98618662828786202550316227938, −3.66394228653786575705186956697, −2.74383360258672917100219227772, −2.42046987814546890332771991388, −1.64852318544201327587218075810, 0,
1.64852318544201327587218075810, 2.42046987814546890332771991388, 2.74383360258672917100219227772, 3.66394228653786575705186956697, 4.98618662828786202550316227938, 5.30143248916480353046201263559, 6.20380229374188543521779153928, 6.67426052034421426303493044464, 7.40931959821777790566999050955
