| L(s) = 1 | − 3-s + 5-s − 4·7-s + 9-s + 13-s − 15-s + 2·17-s + 4·21-s + 25-s − 27-s − 2·29-s + 4·31-s − 4·35-s + 6·37-s − 39-s − 6·41-s − 4·43-s + 45-s + 4·47-s + 9·49-s − 2·51-s − 10·53-s − 2·61-s − 4·63-s + 65-s − 8·67-s − 4·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 0.277·13-s − 0.258·15-s + 0.485·17-s + 0.872·21-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.676·35-s + 0.986·37-s − 0.160·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s + 0.583·47-s + 9/7·49-s − 0.280·51-s − 1.37·53-s − 0.256·61-s − 0.503·63-s + 0.124·65-s − 0.977·67-s − 0.474·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.360952789240966940915034865161, −7.39910808894726191505843423191, −6.60462664522065950803289124773, −6.12468627442412513502239998443, −5.44981292840460319912092670613, −4.44011904166190568895506069501, −3.46536898199853361703547895297, −2.70272459061247762349340288965, −1.33671083067980119322861232528, 0,
1.33671083067980119322861232528, 2.70272459061247762349340288965, 3.46536898199853361703547895297, 4.44011904166190568895506069501, 5.44981292840460319912092670613, 6.12468627442412513502239998443, 6.60462664522065950803289124773, 7.39910808894726191505843423191, 8.360952789240966940915034865161
