| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s + 6·11-s + 12-s + 13-s − 15-s + 16-s − 8·17-s + 18-s − 20-s + 6·22-s + 24-s + 25-s + 26-s + 27-s − 30-s − 10·31-s + 32-s + 6·33-s − 8·34-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.80·11-s + 0.288·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s − 1.94·17-s + 0.235·18-s − 0.223·20-s + 1.27·22-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.182·30-s − 1.79·31-s + 0.176·32-s + 1.04·33-s − 1.37·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−15.93300418424610, −15.23170147852417, −14.90254894187440, −14.35137492450829, −13.89574745172225, −13.27164395211172, −12.83586986657550, −12.17865656926252, −11.60567824318455, −11.15088617545312, −10.68997147177660, −9.697282415733657, −9.218977115470728, −8.634878874283944, −8.213870506362167, −7.174506033710669, −6.804853830609676, −6.426615176611557, −5.482202805333925, −4.692096973582540, −4.095380293007863, −3.662522982094219, −3.010908336689523, −1.908458953735636, −1.521953171699696, 0,
1.521953171699696, 1.908458953735636, 3.010908336689523, 3.662522982094219, 4.095380293007863, 4.692096973582540, 5.482202805333925, 6.426615176611557, 6.804853830609676, 7.174506033710669, 8.213870506362167, 8.634878874283944, 9.218977115470728, 9.697282415733657, 10.68997147177660, 11.15088617545312, 11.60567824318455, 12.17865656926252, 12.83586986657550, 13.27164395211172, 13.89574745172225, 14.35137492450829, 14.90254894187440, 15.23170147852417, 15.93300418424610
