| L(s) = 1 | + 5·3-s − 9·5-s − 2·9-s + 57·11-s − 70·13-s − 45·15-s + 51·17-s − 5·19-s − 69·23-s − 44·25-s − 145·27-s + 114·29-s − 23·31-s + 285·33-s − 253·37-s − 350·39-s − 42·41-s + 124·43-s + 18·45-s − 201·47-s + 255·51-s − 393·53-s − 513·55-s − 25·57-s − 219·59-s − 709·61-s + 630·65-s + ⋯ |
| L(s) = 1 | + 0.962·3-s − 0.804·5-s − 0.0740·9-s + 1.56·11-s − 1.49·13-s − 0.774·15-s + 0.727·17-s − 0.0603·19-s − 0.625·23-s − 0.351·25-s − 1.03·27-s + 0.729·29-s − 0.133·31-s + 1.50·33-s − 1.12·37-s − 1.43·39-s − 0.159·41-s + 0.439·43-s + 0.0596·45-s − 0.623·47-s + 0.700·51-s − 1.01·53-s − 1.25·55-s − 0.0580·57-s − 0.483·59-s − 1.48·61-s + 1.20·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 5 T + p^{3} T^{2} \) |
| 5 | \( 1 + 9 T + p^{3} T^{2} \) |
| 11 | \( 1 - 57 T + p^{3} T^{2} \) |
| 13 | \( 1 + 70 T + p^{3} T^{2} \) |
| 17 | \( 1 - 3 p T + p^{3} T^{2} \) |
| 19 | \( 1 + 5 T + p^{3} T^{2} \) |
| 23 | \( 1 + 3 p T + p^{3} T^{2} \) |
| 29 | \( 1 - 114 T + p^{3} T^{2} \) |
| 31 | \( 1 + 23 T + p^{3} T^{2} \) |
| 37 | \( 1 + 253 T + p^{3} T^{2} \) |
| 41 | \( 1 + 42 T + p^{3} T^{2} \) |
| 43 | \( 1 - 124 T + p^{3} T^{2} \) |
| 47 | \( 1 + 201 T + p^{3} T^{2} \) |
| 53 | \( 1 + 393 T + p^{3} T^{2} \) |
| 59 | \( 1 + 219 T + p^{3} T^{2} \) |
| 61 | \( 1 + 709 T + p^{3} T^{2} \) |
| 67 | \( 1 + 419 T + p^{3} T^{2} \) |
| 71 | \( 1 - 96 T + p^{3} T^{2} \) |
| 73 | \( 1 + 313 T + p^{3} T^{2} \) |
| 79 | \( 1 + 461 T + p^{3} T^{2} \) |
| 83 | \( 1 - 588 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1017 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1834 T + p^{3} T^{2} \) |
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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
−9.396533090684157080841248267913, −8.603234785565696151765672913533, −7.80417655649489616232006206066, −7.14850051034448285516263559707, −6.02738455484485099179178774811, −4.69223601730224531270410607634, −3.78119963812405954827254307426, −2.95996509160900424053183179838, −1.68219201266301967115005417061, 0,
1.68219201266301967115005417061, 2.95996509160900424053183179838, 3.78119963812405954827254307426, 4.69223601730224531270410607634, 6.02738455484485099179178774811, 7.14850051034448285516263559707, 7.80417655649489616232006206066, 8.603234785565696151765672913533, 9.396533090684157080841248267913
