| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 4·7-s + 8-s + 9-s − 11-s + 12-s + 4·14-s + 16-s + 17-s + 18-s − 2·19-s + 4·21-s − 22-s + 24-s − 5·25-s + 27-s + 4·28-s − 6·29-s − 8·31-s + 32-s − 33-s + 34-s + 36-s − 2·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 1.06·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.458·19-s + 0.872·21-s − 0.213·22-s + 0.204·24-s − 25-s + 0.192·27-s + 0.755·28-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.174·33-s + 0.171·34-s + 1/6·36-s − 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.712983738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.712983738\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.08142626709802, −12.77578753167145, −12.18477990243577, −11.80403846408051, −11.12188994868082, −10.94086836490262, −10.50992187736094, −9.751407575471143, −9.319507084126064, −8.736395577506440, −8.259676665957555, −7.769529193229499, −7.378421346652463, −7.075561918378156, −6.083802212588082, −5.754269734718197, −5.213038085257280, −4.740638070291137, −4.111947321591945, −3.796103214169029, −3.161256344316039, −2.265499236703333, −2.057644695847902, −1.488525943375051, −0.5950829907749451,
0.5950829907749451, 1.488525943375051, 2.057644695847902, 2.265499236703333, 3.161256344316039, 3.796103214169029, 4.111947321591945, 4.740638070291137, 5.213038085257280, 5.754269734718197, 6.083802212588082, 7.075561918378156, 7.378421346652463, 7.769529193229499, 8.259676665957555, 8.736395577506440, 9.319507084126064, 9.751407575471143, 10.50992187736094, 10.94086836490262, 11.12188994868082, 11.80403846408051, 12.18477990243577, 12.77578753167145, 13.08142626709802
