| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s + 6·11-s − 12-s + 2·13-s − 15-s + 16-s − 18-s − 6·19-s + 20-s − 6·22-s − 23-s + 24-s + 25-s − 2·26-s − 27-s − 4·29-s + 30-s − 2·31-s − 32-s − 6·33-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 + 0.554·13-s − 0.258·15-s + 1/4·16-s − 0.235·18-s − 1.37·19-s + 0.223·20-s − 1.27·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.742·29-s + 0.182·30-s − 0.359·31-s − 0.176·32-s − 1.04·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−15.33319006290090, −14.75924711910402, −14.27336339386906, −13.74228031171170, −13.02939474417954, −12.48011150407109, −12.03965619900161, −11.40287874997149, −11.00692236130717, −10.50091456507779, −9.882549444310047, −9.321709953103828, −8.869075877022975, −8.432539524692693, −7.620158496002260, −6.909442742746575, −6.466044661022709, −6.129166153649075, −5.429739149748891, −4.623829908040103, −3.876722262971397, −3.451970382336806, −2.225836429830247, −1.694855618313872, −1.037356216242678, 0,
1.037356216242678, 1.694855618313872, 2.225836429830247, 3.451970382336806, 3.876722262971397, 4.623829908040103, 5.429739149748891, 6.129166153649075, 6.466044661022709, 6.909442742746575, 7.620158496002260, 8.432539524692693, 8.869075877022975, 9.321709953103828, 9.882549444310047, 10.50091456507779, 11.00692236130717, 11.40287874997149, 12.03965619900161, 12.48011150407109, 13.02939474417954, 13.74228031171170, 14.27336339386906, 14.75924711910402, 15.33319006290090
