| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s + 3·11-s − 12-s − 6·13-s + 14-s − 15-s + 16-s − 18-s + 4·19-s + 20-s + 21-s − 3·22-s + 24-s + 25-s + 6·26-s − 27-s − 28-s + 3·29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.904·11-s − 0.288·12-s − 1.66·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.218·21-s − 0.639·22-s + 0.204·24-s + 1/5·25-s + 1.17·26-s − 0.192·27-s − 0.188·28-s + 0.557·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 11 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
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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
−12.54375626943712, −12.11359049666251, −11.79797516826545, −11.52976368737925, −10.68736277886255, −10.45133584988387, −9.922573288080108, −9.635595225633702, −9.190124820862603, −8.809736781640759, −8.142148416416694, −7.590843529796220, −7.158494690443209, −6.758999065739023, −6.425322663097569, −5.745406018861179, −5.371553993511722, −4.778928167034099, −4.409598373483452, −3.496050910370586, −3.160377626989505, −2.456746538335723, −1.888355570034879, −1.346902151349081, −0.6624211578177082, 0,
0.6624211578177082, 1.346902151349081, 1.888355570034879, 2.456746538335723, 3.160377626989505, 3.496050910370586, 4.409598373483452, 4.778928167034099, 5.371553993511722, 5.745406018861179, 6.425322663097569, 6.758999065739023, 7.158494690443209, 7.590843529796220, 8.142148416416694, 8.809736781640759, 9.190124820862603, 9.635595225633702, 9.922573288080108, 10.45133584988387, 10.68736277886255, 11.52976368737925, 11.79797516826545, 12.11359049666251, 12.54375626943712
