| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 3·7-s + 8-s + 9-s + 11-s + 12-s − 13-s + 3·14-s + 16-s + 17-s + 18-s − 4·19-s + 3·21-s + 22-s − 2·23-s + 24-s − 26-s + 27-s + 3·28-s − 6·29-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 + 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s − 0.277·13-s + 0.801·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.917·19-s + 0.654·21-s + 0.213·22-s − 0.417·23-s + 0.204·24-s − 0.196·26-s + 0.192·27-s + 0.566·28-s − 1.11·29-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 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 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
−12.76009220417158, −12.40127326007557, −11.82060938579121, −11.46814536862623, −10.94356128324975, −10.70715598499390, −9.914574049052447, −9.726115981266655, −8.981672606073329, −8.522876619441170, −8.212860053539706, −7.636117503605049, −7.273468499264041, −6.746827014641504, −6.263122726929922, −5.530262801899431, −5.285116345463835, −4.702465097035267, −4.149969134284107, −3.812951412438189, −3.289003593436900, −2.550598636991610, −1.980183056579891, −1.758862279585805, −0.9835434608319433, 0,
0.9835434608319433, 1.758862279585805, 1.980183056579891, 2.550598636991610, 3.289003593436900, 3.812951412438189, 4.149969134284107, 4.702465097035267, 5.285116345463835, 5.530262801899431, 6.263122726929922, 6.746827014641504, 7.273468499264041, 7.636117503605049, 8.212860053539706, 8.522876619441170, 8.981672606073329, 9.726115981266655, 9.914574049052447, 10.70715598499390, 10.94356128324975, 11.46814536862623, 11.82060938579121, 12.40127326007557, 12.76009220417158
