| L(s) = 1 | − 3-s + 2·5-s − 7-s + 9-s − 4·11-s − 6·13-s − 2·15-s + 2·17-s − 19-s + 21-s − 4·23-s − 25-s − 27-s + 6·29-s + 4·31-s + 4·33-s − 2·35-s + 2·37-s + 6·39-s + 10·41-s + 12·43-s + 2·45-s + 4·47-s + 49-s − 2·51-s + 14·53-s − 8·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.66·13-s − 0.516·15-s + 0.485·17-s − 0.229·19-s + 0.218·21-s − 0.834·23-s − 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.718·31-s + 0.696·33-s − 0.338·35-s + 0.328·37-s + 0.960·39-s + 1.56·41-s + 1.82·43-s + 0.298·45-s + 0.583·47-s + 1/7·49-s − 0.280·51-s + 1.92·53-s − 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.282027082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.282027082\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−8.748252260970204212799564412576, −7.66121026502410780453416848082, −7.30284570347130335207977472549, −6.14072694581602486414824019299, −5.75197877543014727546330841866, −4.95080815681984432602405164076, −4.19106295133619833106533954620, −2.69446415603566274703281560766, −2.27996871670795650855846913109, −0.67341498658754368439450011463,
0.67341498658754368439450011463, 2.27996871670795650855846913109, 2.69446415603566274703281560766, 4.19106295133619833106533954620, 4.95080815681984432602405164076, 5.75197877543014727546330841866, 6.14072694581602486414824019299, 7.30284570347130335207977472549, 7.66121026502410780453416848082, 8.748252260970204212799564412576
