L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 5·7-s − 8-s + 9-s + 11-s + 12-s − 13-s − 5·14-s + 16-s + 17-s − 18-s − 2·19-s + 5·21-s − 22-s − 6·23-s − 24-s + 26-s + 27-s + 5·28-s − 9·29-s − 4·31-s − 32-s + 33-s − 34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.88·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s − 0.277·13-s − 1.33·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.458·19-s + 1.09·21-s − 0.213·22-s − 1.25·23-s − 0.204·24-s + 0.196·26-s + 0.192·27-s + 0.944·28-s − 1.67·29-s − 0.718·31-s − 0.176·32-s + 0.174·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.69662882689257, −12.23099855565321, −11.58550991976146, −11.36426056827178, −11.02562311418279, −10.36554948613188, −10.06994455925170, −9.496894841309464, −8.904928577891626, −8.735595334746856, −8.118636100235843, −7.809652171142442, −7.379696406332859, −7.091905737946121, −6.248146485736833, −5.728009793272378, −5.349798406176489, −4.644673748570115, −4.219410614944867, −3.747188954918908, −3.089919982724317, −2.279684779699735, −1.879655009435477, −1.642302608572561, −0.8578836176088905, 0,
0.8578836176088905, 1.642302608572561, 1.879655009435477, 2.279684779699735, 3.089919982724317, 3.747188954918908, 4.219410614944867, 4.644673748570115, 5.349798406176489, 5.728009793272378, 6.248146485736833, 7.091905737946121, 7.379696406332859, 7.809652171142442, 8.118636100235843, 8.735595334746856, 8.904928577891626, 9.496894841309464, 10.06994455925170, 10.36554948613188, 11.02562311418279, 11.36426056827178, 11.58550991976146, 12.23099855565321, 12.69662882689257