| L(s) = 1 | − 2.70·3-s − 2.61·5-s − 1.43·7-s + 4.33·9-s − 3.64·11-s − 2.91·13-s + 7.07·15-s + 0.0204·17-s + 1.79·19-s + 3.87·21-s + 2.66·23-s + 1.82·25-s − 3.61·27-s + 6.24·29-s + 4.99·31-s + 9.88·33-s + 3.73·35-s + 0.458·37-s + 7.90·39-s − 1.83·41-s − 0.880·43-s − 11.3·45-s + 9.50·47-s − 4.95·49-s − 0.0553·51-s + 2.12·53-s + 9.53·55-s + ⋯ |
| L(s) = 1 | − 1.56·3-s − 1.16·5-s − 0.540·7-s + 1.44·9-s − 1.10·11-s − 0.809·13-s + 1.82·15-s + 0.00495·17-s + 0.411·19-s + 0.845·21-s + 0.555·23-s + 0.364·25-s − 0.695·27-s + 1.16·29-s + 0.897·31-s + 1.72·33-s + 0.631·35-s + 0.0753·37-s + 1.26·39-s − 0.286·41-s − 0.134·43-s − 1.68·45-s + 1.38·47-s − 0.707·49-s − 0.00775·51-s + 0.292·53-s + 1.28·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 + 2.70T + 3T^{2} \) |
| 5 | \( 1 + 2.61T + 5T^{2} \) |
| 7 | \( 1 + 1.43T + 7T^{2} \) |
| 11 | \( 1 + 3.64T + 11T^{2} \) |
| 13 | \( 1 + 2.91T + 13T^{2} \) |
| 17 | \( 1 - 0.0204T + 17T^{2} \) |
| 19 | \( 1 - 1.79T + 19T^{2} \) |
| 23 | \( 1 - 2.66T + 23T^{2} \) |
| 29 | \( 1 - 6.24T + 29T^{2} \) |
| 31 | \( 1 - 4.99T + 31T^{2} \) |
| 37 | \( 1 - 0.458T + 37T^{2} \) |
| 41 | \( 1 + 1.83T + 41T^{2} \) |
| 43 | \( 1 + 0.880T + 43T^{2} \) |
| 47 | \( 1 - 9.50T + 47T^{2} \) |
| 53 | \( 1 - 2.12T + 53T^{2} \) |
| 59 | \( 1 - 1.13T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 - 4.26T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 - 8.58T + 79T^{2} \) |
| 83 | \( 1 + 5.09T + 83T^{2} \) |
| 89 | \( 1 - 2.58T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−7.80513827043526506015460400769, −7.33055784487495483498693766249, −6.56406458833107588634097933362, −5.86720581724574760933989152946, −4.91309985527412242587238497482, −4.66625554961966364102950547233, −3.50070609781604134514587559619, −2.58435521603116061701347941452, −0.848840689806556500712799689157, 0,
0.848840689806556500712799689157, 2.58435521603116061701347941452, 3.50070609781604134514587559619, 4.66625554961966364102950547233, 4.91309985527412242587238497482, 5.86720581724574760933989152946, 6.56406458833107588634097933362, 7.33055784487495483498693766249, 7.80513827043526506015460400769
