| L(s) = 1 | − 3-s − 7-s − 2·9-s − 2·11-s + 3·13-s + 21-s + 23-s − 5·25-s + 5·27-s − 29-s + 5·31-s + 2·33-s + 8·37-s − 3·39-s − 7·41-s − 4·43-s − 3·47-s + 49-s + 12·53-s + 4·59-s + 6·61-s + 2·63-s − 12·67-s − 69-s − 13·71-s + 3·73-s + 5·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s − 2/3·9-s − 0.603·11-s + 0.832·13-s + 0.218·21-s + 0.208·23-s − 25-s + 0.962·27-s − 0.185·29-s + 0.898·31-s + 0.348·33-s + 1.31·37-s − 0.480·39-s − 1.09·41-s − 0.609·43-s − 0.437·47-s + 1/7·49-s + 1.64·53-s + 0.520·59-s + 0.768·61-s + 0.251·63-s − 1.46·67-s − 0.120·69-s − 1.54·71-s + 0.351·73-s + 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−16.88941970555024, −16.20987572480183, −16.00109354151029, −15.08940057549862, −14.79952695128219, −13.83612344684908, −13.35159653327621, −13.04959841205039, −11.95430010535234, −11.79637211272817, −11.12115283362954, −10.43579691080432, −10.02609568378574, −9.191950284488579, −8.528409379846883, −8.032139802988622, −7.235426705068892, −6.420446275140239, −5.977697709508441, −5.371035807798221, −4.634589286947621, −3.750589003646881, −3.021407998297186, −2.217884398614673, −1.016075322856445, 0,
1.016075322856445, 2.217884398614673, 3.021407998297186, 3.750589003646881, 4.634589286947621, 5.371035807798221, 5.977697709508441, 6.420446275140239, 7.235426705068892, 8.032139802988622, 8.528409379846883, 9.191950284488579, 10.02609568378574, 10.43579691080432, 11.12115283362954, 11.79637211272817, 11.95430010535234, 13.04959841205039, 13.35159653327621, 13.83612344684908, 14.79952695128219, 15.08940057549862, 16.00109354151029, 16.20987572480183, 16.88941970555024
