| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 2·7-s − 8-s + 9-s − 10-s + 11-s + 12-s − 4·13-s − 2·14-s + 15-s + 16-s + 17-s − 18-s − 8·19-s + 20-s + 2·21-s − 22-s − 8·23-s − 24-s + 25-s + 4·26-s + 27-s + 2·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s − 1.10·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 1.83·19-s + 0.223·20-s + 0.436·21-s − 0.213·22-s − 1.66·23-s − 0.204·24-s + 1/5·25-s + 0.784·26-s + 0.192·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.81138900093119154787799404520, −7.42523426001964815320872117056, −6.34353548247863440434081257161, −5.90692833774843483744258589430, −4.68053237396923956700748176412, −4.19800884107099726749620369319, −2.95155828848732508029863535787, −2.11943837372233247320900528031, −1.58975997438997466453997279037, 0,
1.58975997438997466453997279037, 2.11943837372233247320900528031, 2.95155828848732508029863535787, 4.19800884107099726749620369319, 4.68053237396923956700748176412, 5.90692833774843483744258589430, 6.34353548247863440434081257161, 7.42523426001964815320872117056, 7.81138900093119154787799404520
