| L(s) = 1 | − 2·2-s + 4·4-s − 6·5-s + 7·7-s − 8·8-s + 12·10-s − 30·11-s + 2·13-s − 14·14-s + 16·16-s − 66·17-s − 52·19-s − 24·20-s + 60·22-s − 114·23-s − 89·25-s − 4·26-s + 28·28-s − 72·29-s − 196·31-s − 32·32-s + 132·34-s − 42·35-s − 286·37-s + 104·38-s + 48·40-s + 378·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.536·5-s + 0.377·7-s − 0.353·8-s + 0.379·10-s − 0.822·11-s + 0.0426·13-s − 0.267·14-s + 1/4·16-s − 0.941·17-s − 0.627·19-s − 0.268·20-s + 0.581·22-s − 1.03·23-s − 0.711·25-s − 0.0301·26-s + 0.188·28-s − 0.461·29-s − 1.13·31-s − 0.176·32-s + 0.665·34-s − 0.202·35-s − 1.27·37-s + 0.443·38-s + 0.189·40-s + 1.43·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 5 | \( 1 + 6 T + p^{3} T^{2} \) |
| 11 | \( 1 + 30 T + p^{3} T^{2} \) |
| 13 | \( 1 - 2 T + p^{3} T^{2} \) |
| 17 | \( 1 + 66 T + p^{3} T^{2} \) |
| 19 | \( 1 + 52 T + p^{3} T^{2} \) |
| 23 | \( 1 + 114 T + p^{3} T^{2} \) |
| 29 | \( 1 + 72 T + p^{3} T^{2} \) |
| 31 | \( 1 + 196 T + p^{3} T^{2} \) |
| 37 | \( 1 + 286 T + p^{3} T^{2} \) |
| 41 | \( 1 - 378 T + p^{3} T^{2} \) |
| 43 | \( 1 - 164 T + p^{3} T^{2} \) |
| 47 | \( 1 - 228 T + p^{3} T^{2} \) |
| 53 | \( 1 - 348 T + p^{3} T^{2} \) |
| 59 | \( 1 - 348 T + p^{3} T^{2} \) |
| 61 | \( 1 + 106 T + p^{3} T^{2} \) |
| 67 | \( 1 - 596 T + p^{3} T^{2} \) |
| 71 | \( 1 + 630 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1042 T + p^{3} T^{2} \) |
| 79 | \( 1 + 88 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1440 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1374 T + p^{3} T^{2} \) |
| 97 | \( 1 + 34 T + p^{3} T^{2} \) |
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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
−12.20315377968678251272493456515, −11.16624149317558385604020606624, −10.40248815949350552212426085696, −9.077669375375997723559780642491, −8.095997948509729871143460776860, −7.20029494437065438986456466884, −5.70859343306553849986402970420, −4.06833298161340735277736179315, −2.18148648782316378584970915932, 0,
2.18148648782316378584970915932, 4.06833298161340735277736179315, 5.70859343306553849986402970420, 7.20029494437065438986456466884, 8.095997948509729871143460776860, 9.077669375375997723559780642491, 10.40248815949350552212426085696, 11.16624149317558385604020606624, 12.20315377968678251272493456515
