| L(s) = 1 | − 3-s + 7-s + 9-s + 11-s − 13-s + 5·17-s − 2·19-s − 21-s − 7·23-s − 27-s + 4·31-s − 33-s − 7·37-s + 39-s − 11·41-s + 6·43-s − 6·49-s − 5·51-s + 11·53-s + 2·57-s − 4·59-s + 7·61-s + 63-s + 8·67-s + 7·69-s − 9·71-s − 2·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s + 1.21·17-s − 0.458·19-s − 0.218·21-s − 1.45·23-s − 0.192·27-s + 0.718·31-s − 0.174·33-s − 1.15·37-s + 0.160·39-s − 1.71·41-s + 0.914·43-s − 6/7·49-s − 0.700·51-s + 1.51·53-s + 0.264·57-s − 0.520·59-s + 0.896·61-s + 0.125·63-s + 0.977·67-s + 0.842·69-s − 1.06·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.51627125623879, −14.02034853778487, −13.59961106324018, −12.92616021061087, −12.32636794477769, −11.94398618493718, −11.69138386490129, −10.99003987321344, −10.28001931133466, −10.14524362858829, −9.564131531957951, −8.785749676740294, −8.281741986587729, −7.833626052896646, −7.163326862090365, −6.687362134758070, −6.031042137718830, −5.562253727864293, −5.012651114973692, −4.400586481563409, −3.781352940365655, −3.215647017636776, −2.253820563871345, −1.689098010182182, −0.9058240692903022, 0,
0.9058240692903022, 1.689098010182182, 2.253820563871345, 3.215647017636776, 3.781352940365655, 4.400586481563409, 5.012651114973692, 5.562253727864293, 6.031042137718830, 6.687362134758070, 7.163326862090365, 7.833626052896646, 8.281741986587729, 8.785749676740294, 9.564131531957951, 10.14524362858829, 10.28001931133466, 10.99003987321344, 11.69138386490129, 11.94398618493718, 12.32636794477769, 12.92616021061087, 13.59961106324018, 14.02034853778487, 14.51627125623879
