| L(s) = 1 | − 2-s − 4-s − 5-s + 7-s + 3·8-s + 10-s − 5·11-s − 5·13-s − 14-s − 16-s − 4·17-s + 4·19-s + 20-s + 5·22-s − 6·23-s − 4·25-s + 5·26-s − 28-s − 7·31-s − 5·32-s + 4·34-s − 35-s + 10·37-s − 4·38-s − 3·40-s + 9·43-s + 5·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 0.377·7-s + 1.06·8-s + 0.316·10-s − 1.50·11-s − 1.38·13-s − 0.267·14-s − 1/4·16-s − 0.970·17-s + 0.917·19-s + 0.223·20-s + 1.06·22-s − 1.25·23-s − 4/5·25-s + 0.980·26-s − 0.188·28-s − 1.25·31-s − 0.883·32-s + 0.685·34-s − 0.169·35-s + 1.64·37-s − 0.648·38-s − 0.474·40-s + 1.37·43-s + 0.753·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−14.64405870513340, −14.19971969106819, −13.69165744508117, −13.20938638792869, −12.64318866487960, −12.20301832795707, −11.55979304923198, −10.93845594726759, −10.58677941507704, −9.992697865791910, −9.429999334163232, −9.188079653899747, −8.309499715000679, −7.865925270783385, −7.488131443626066, −7.291551034221545, −6.114676684837601, −5.519907914073536, −5.017860479016564, −4.352656885056173, −4.060259271730131, −2.980493982993558, −2.345519674763761, −1.754326840549449, −0.5879278435933712, 0,
0.5879278435933712, 1.754326840549449, 2.345519674763761, 2.980493982993558, 4.060259271730131, 4.352656885056173, 5.017860479016564, 5.519907914073536, 6.114676684837601, 7.291551034221545, 7.488131443626066, 7.865925270783385, 8.309499715000679, 9.188079653899747, 9.429999334163232, 9.992697865791910, 10.58677941507704, 10.93845594726759, 11.55979304923198, 12.20301832795707, 12.64318866487960, 13.20938638792869, 13.69165744508117, 14.19971969106819, 14.64405870513340
