| L(s) = 1 | − 2·3-s − 2·5-s + 2·7-s + 9-s − 6·11-s − 2·13-s + 4·15-s − 4·21-s − 6·23-s − 25-s + 4·27-s − 10·29-s − 2·31-s + 12·33-s − 4·35-s + 6·37-s + 4·39-s + 6·41-s + 8·43-s − 2·45-s − 3·49-s + 10·53-s + 12·55-s + 8·59-s + 14·61-s + 2·63-s + 4·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 0.755·7-s + 1/3·9-s − 1.80·11-s − 0.554·13-s + 1.03·15-s − 0.872·21-s − 1.25·23-s − 1/5·25-s + 0.769·27-s − 1.85·29-s − 0.359·31-s + 2.08·33-s − 0.676·35-s + 0.986·37-s + 0.640·39-s + 0.937·41-s + 1.21·43-s − 0.298·45-s − 3/7·49-s + 1.37·53-s + 1.61·55-s + 1.04·59-s + 1.79·61-s + 0.251·63-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−16.12782247613765, −15.67800038387927, −14.90374673474050, −14.66155598686821, −13.80316389859248, −13.11955503894726, −12.60927836642497, −12.10585235705614, −11.51639643329274, −11.04604834979821, −10.81185887762455, −9.999729609583474, −9.492711693073930, −8.390566961522212, −8.028815394744493, −7.504638200050431, −7.035801430831533, −5.917355747674808, −5.559094468889970, −5.096281122619478, −4.304216076799834, −3.804416992059989, −2.624037456021498, −2.069094674802884, −0.6961193999450161, 0,
0.6961193999450161, 2.069094674802884, 2.624037456021498, 3.804416992059989, 4.304216076799834, 5.096281122619478, 5.559094468889970, 5.917355747674808, 7.035801430831533, 7.504638200050431, 8.028815394744493, 8.390566961522212, 9.492711693073930, 9.999729609583474, 10.81185887762455, 11.04604834979821, 11.51639643329274, 12.10585235705614, 12.60927836642497, 13.11955503894726, 13.80316389859248, 14.66155598686821, 14.90374673474050, 15.67800038387927, 16.12782247613765
