| L(s) = 1 | + 3-s + 5-s − 7-s − 2·9-s − 3·11-s − 3·13-s + 15-s + 3·17-s − 21-s − 23-s + 25-s − 5·27-s + 9·29-s − 2·31-s − 3·33-s − 35-s + 4·37-s − 3·39-s + 4·41-s − 2·43-s − 2·45-s − 13·47-s + 49-s + 3·51-s + 2·53-s − 3·55-s − 6·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s − 2/3·9-s − 0.904·11-s − 0.832·13-s + 0.258·15-s + 0.727·17-s − 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.962·27-s + 1.67·29-s − 0.359·31-s − 0.522·33-s − 0.169·35-s + 0.657·37-s − 0.480·39-s + 0.624·41-s − 0.304·43-s − 0.298·45-s − 1.89·47-s + 1/7·49-s + 0.420·51-s + 0.274·53-s − 0.404·55-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−14.57551572594352, −14.35740902977902, −13.69499509083559, −13.31672421608982, −12.76771533118714, −12.24398651683733, −11.75113925172337, −11.14206409433557, −10.35868137493496, −10.17717433203421, −9.484206990959251, −9.140989674209019, −8.383608092142609, −7.947398343481629, −7.559364747079243, −6.764411603569273, −6.180527441740243, −5.677723249209817, −4.978275455110620, −4.612676935780419, −3.515729472588248, −3.112875419942913, −2.501445217254918, −2.020529609114020, −0.9205415544501905, 0,
0.9205415544501905, 2.020529609114020, 2.501445217254918, 3.112875419942913, 3.515729472588248, 4.612676935780419, 4.978275455110620, 5.677723249209817, 6.180527441740243, 6.764411603569273, 7.559364747079243, 7.947398343481629, 8.383608092142609, 9.140989674209019, 9.484206990959251, 10.17717433203421, 10.35868137493496, 11.14206409433557, 11.75113925172337, 12.24398651683733, 12.76771533118714, 13.31672421608982, 13.69499509083559, 14.35740902977902, 14.57551572594352
