| L(s) = 1 | + 3·3-s − 10·5-s − 7·7-s + 9·9-s + 12·11-s + 30·13-s − 30·15-s + 34·17-s − 148·19-s − 21·21-s − 152·23-s − 25·25-s + 27·27-s − 106·29-s − 304·31-s + 36·33-s + 70·35-s − 114·37-s + 90·39-s + 202·41-s − 116·43-s − 90·45-s − 224·47-s + 49·49-s + 102·51-s − 274·53-s − 120·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.328·11-s + 0.640·13-s − 0.516·15-s + 0.485·17-s − 1.78·19-s − 0.218·21-s − 1.37·23-s − 1/5·25-s + 0.192·27-s − 0.678·29-s − 1.76·31-s + 0.189·33-s + 0.338·35-s − 0.506·37-s + 0.369·39-s + 0.769·41-s − 0.411·43-s − 0.298·45-s − 0.695·47-s + 1/7·49-s + 0.280·51-s − 0.710·53-s − 0.294·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 + p T \) |
| good | 5 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 30 T + p^{3} T^{2} \) |
| 17 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 19 | \( 1 + 148 T + p^{3} T^{2} \) |
| 23 | \( 1 + 152 T + p^{3} T^{2} \) |
| 29 | \( 1 + 106 T + p^{3} T^{2} \) |
| 31 | \( 1 + 304 T + p^{3} T^{2} \) |
| 37 | \( 1 + 114 T + p^{3} T^{2} \) |
| 41 | \( 1 - 202 T + p^{3} T^{2} \) |
| 43 | \( 1 + 116 T + p^{3} T^{2} \) |
| 47 | \( 1 + 224 T + p^{3} T^{2} \) |
| 53 | \( 1 + 274 T + p^{3} T^{2} \) |
| 59 | \( 1 - 660 T + p^{3} T^{2} \) |
| 61 | \( 1 - 382 T + p^{3} T^{2} \) |
| 67 | \( 1 + 12 T + p^{3} T^{2} \) |
| 71 | \( 1 - 552 T + p^{3} T^{2} \) |
| 73 | \( 1 + 614 T + p^{3} T^{2} \) |
| 79 | \( 1 + 880 T + p^{3} T^{2} \) |
| 83 | \( 1 - 108 T + p^{3} T^{2} \) |
| 89 | \( 1 + 86 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1426 T + p^{3} T^{2} \) |
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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
−10.69016277315891233656152665294, −9.657937804193542364900762643509, −8.638885954802964202956600690123, −7.955765069573701785111334525963, −6.92015590059259223638448872427, −5.81917927383298404226699125277, −4.14588315318691583224823160744, −3.57215610325400514085082382261, −1.94413213541946236781267209249, 0,
1.94413213541946236781267209249, 3.57215610325400514085082382261, 4.14588315318691583224823160744, 5.81917927383298404226699125277, 6.92015590059259223638448872427, 7.955765069573701785111334525963, 8.638885954802964202956600690123, 9.657937804193542364900762643509, 10.69016277315891233656152665294
