| L(s) = 1 | − 3-s + 5-s − 3·7-s + 9-s − 15-s − 17-s − 4·19-s + 3·21-s + 25-s − 27-s − 2·29-s − 3·35-s + 8·37-s + 2·41-s + 8·43-s + 45-s + 8·47-s + 2·49-s + 51-s − 4·53-s + 4·57-s − 5·59-s + 13·61-s − 3·63-s + 7·67-s + 9·71-s + 6·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.13·7-s + 1/3·9-s − 0.258·15-s − 0.242·17-s − 0.917·19-s + 0.654·21-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.507·35-s + 1.31·37-s + 0.312·41-s + 1.21·43-s + 0.149·45-s + 1.16·47-s + 2/7·49-s + 0.140·51-s − 0.549·53-s + 0.529·57-s − 0.650·59-s + 1.66·61-s − 0.377·63-s + 0.855·67-s + 1.06·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.903859063\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.903859063\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−12.63999975106986, −12.30749322473077, −11.56458573429519, −11.18242434671491, −10.70401097952269, −10.39359360107432, −9.739544504318051, −9.470799011774338, −9.098506328208991, −8.504473597365911, −7.877637748452422, −7.466827448925212, −6.791794710832896, −6.455228886221145, −6.124990034367563, −5.645002097716717, −5.087139488466365, −4.560878770039661, −3.838741645847735, −3.703693043408430, −2.686976590699538, −2.450073801939393, −1.778650258570174, −0.8633610344108579, −0.4692010240902838,
0.4692010240902838, 0.8633610344108579, 1.778650258570174, 2.450073801939393, 2.686976590699538, 3.703693043408430, 3.838741645847735, 4.560878770039661, 5.087139488466365, 5.645002097716717, 6.124990034367563, 6.455228886221145, 6.791794710832896, 7.466827448925212, 7.877637748452422, 8.504473597365911, 9.098506328208991, 9.470799011774338, 9.739544504318051, 10.39359360107432, 10.70401097952269, 11.18242434671491, 11.56458573429519, 12.30749322473077, 12.63999975106986
