| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 2·7-s − 8-s + 9-s + 10-s + 6·11-s + 12-s − 6·13-s − 2·14-s − 15-s + 16-s − 18-s − 19-s − 20-s + 2·21-s − 6·22-s + 6·23-s − 24-s + 25-s + 6·26-s + 27-s + 2·28-s + 4·29-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 + 1.80·11-s + 0.288·12-s − 1.66·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.229·19-s − 0.223·20-s + 0.436·21-s − 1.27·22-s + 1.25·23-s − 0.204·24-s + 1/5·25-s + 1.17·26-s + 0.192·27-s + 0.377·28-s + 0.742·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.148531646\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.148531646\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−13.09307732233564, −12.63639773068394, −12.30443123682826, −11.67512013116025, −11.37067678706229, −10.96581648909841, −10.31606863185360, −9.683184215544403, −9.397537557423686, −8.979293796343595, −8.486255296779164, −7.939943660375856, −7.549562096182050, −6.964546521893434, −6.734854884915926, −6.065959114066096, −5.145401564231714, −4.827640582870723, −4.172107333955998, −3.679479520978786, −3.023377150016528, −2.314188299608105, −1.923624270811272, −1.063531672104393, −0.6478320959991266,
0.6478320959991266, 1.063531672104393, 1.923624270811272, 2.314188299608105, 3.023377150016528, 3.679479520978786, 4.172107333955998, 4.827640582870723, 5.145401564231714, 6.065959114066096, 6.734854884915926, 6.964546521893434, 7.549562096182050, 7.939943660375856, 8.486255296779164, 8.979293796343595, 9.397537557423686, 9.683184215544403, 10.31606863185360, 10.96581648909841, 11.37067678706229, 11.67512013116025, 12.30443123682826, 12.63639773068394, 13.09307732233564
