L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s − 3·11-s + 12-s − 5·13-s + 14-s + 16-s + 5·17-s − 18-s − 21-s + 3·22-s − 4·23-s − 24-s + 5·26-s + 27-s − 28-s + 6·29-s − 2·31-s − 32-s − 3·33-s − 5·34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.904·11-s + 0.288·12-s − 1.38·13-s + 0.267·14-s + 1/4·16-s + 1.21·17-s − 0.235·18-s − 0.218·21-s + 0.639·22-s − 0.834·23-s − 0.204·24-s + 0.980·26-s + 0.192·27-s − 0.188·28-s + 1.11·29-s − 0.359·31-s − 0.176·32-s − 0.522·33-s − 0.857·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−12.72842974133603, −12.11866191140125, −11.94239922613098, −11.36988238804563, −10.62053466116204, −10.28957934936075, −9.966203089796776, −9.621546848819449, −9.171533122838707, −8.537339100924350, −8.070527501698591, −7.832806190077918, −7.284976131799119, −6.972426946653267, −6.310630931790035, −5.786591908106605, −5.229609971402972, −4.844962149731165, −4.087251616985692, −3.602039943533575, −2.875836542079740, −2.615585735354164, −2.132933653014409, −1.373588276893216, −0.6808960556481105, 0,
0.6808960556481105, 1.373588276893216, 2.132933653014409, 2.615585735354164, 2.875836542079740, 3.602039943533575, 4.087251616985692, 4.844962149731165, 5.229609971402972, 5.786591908106605, 6.310630931790035, 6.972426946653267, 7.284976131799119, 7.832806190077918, 8.070527501698591, 8.537339100924350, 9.171533122838707, 9.621546848819449, 9.966203089796776, 10.28957934936075, 10.62053466116204, 11.36988238804563, 11.94239922613098, 12.11866191140125, 12.72842974133603