| L(s) = 1 | − 3·3-s + 5-s + 7-s + 6·9-s − 6·11-s + 5·13-s − 3·15-s − 2·17-s − 2·19-s − 3·21-s + 23-s + 25-s − 9·27-s − 29-s + 9·31-s + 18·33-s + 35-s + 12·37-s − 15·39-s + 41-s + 2·43-s + 6·45-s + 7·47-s + 49-s + 6·51-s + 4·53-s − 6·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.447·5-s + 0.377·7-s + 2·9-s − 1.80·11-s + 1.38·13-s − 0.774·15-s − 0.485·17-s − 0.458·19-s − 0.654·21-s + 0.208·23-s + 1/5·25-s − 1.73·27-s − 0.185·29-s + 1.61·31-s + 3.13·33-s + 0.169·35-s + 1.97·37-s − 2.40·39-s + 0.156·41-s + 0.304·43-s + 0.894·45-s + 1.02·47-s + 1/7·49-s + 0.840·51-s + 0.549·53-s − 0.809·55-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 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−15.02421935794659, −14.01133000594108, −13.51742925065734, −13.13885976910809, −12.73719107907680, −12.14973740673980, −11.51714987212173, −11.07503865304396, −10.72522233286409, −10.33577563093575, −9.860814612212648, −9.002417203705118, −8.477812415927675, −7.727822064461186, −7.393581368576385, −6.471196566101526, −6.079114195930713, −5.776443074225251, −5.156024539292699, −4.438974677347467, −4.313442301629986, −3.016919494185863, −2.443549127901458, −1.476690740202209, −0.8625606779556347, 0,
0.8625606779556347, 1.476690740202209, 2.443549127901458, 3.016919494185863, 4.313442301629986, 4.438974677347467, 5.156024539292699, 5.776443074225251, 6.079114195930713, 6.471196566101526, 7.393581368576385, 7.727822064461186, 8.477812415927675, 9.002417203705118, 9.860814612212648, 10.33577563093575, 10.72522233286409, 11.07503865304396, 11.51714987212173, 12.14973740673980, 12.73719107907680, 13.13885976910809, 13.51742925065734, 14.01133000594108, 15.02421935794659
