| L(s) = 1 | + 3-s − 7-s + 9-s − 11-s − 2·13-s − 3·17-s − 19-s − 21-s − 7·23-s + 27-s + 3·29-s + 4·31-s − 33-s + 8·37-s − 2·39-s + 6·41-s + 7·43-s + 10·47-s + 49-s − 3·51-s − 3·53-s − 57-s + 9·59-s − 9·61-s − 63-s − 12·67-s − 7·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.727·17-s − 0.229·19-s − 0.218·21-s − 1.45·23-s + 0.192·27-s + 0.557·29-s + 0.718·31-s − 0.174·33-s + 1.31·37-s − 0.320·39-s + 0.937·41-s + 1.06·43-s + 1.45·47-s + 1/7·49-s − 0.420·51-s − 0.412·53-s − 0.132·57-s + 1.17·59-s − 1.15·61-s − 0.125·63-s − 1.46·67-s − 0.842·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.768936729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.768936729\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−12.50866831685527, −12.12272969205445, −11.72801877163942, −10.97415553724217, −10.76051664150381, −10.08337044349851, −9.821412838703098, −9.355946871656299, −8.837781422743307, −8.453190966304024, −7.866163469723923, −7.530709056425783, −7.104783899826098, −6.424150791019623, −5.989172012548128, −5.718329572009714, −4.677559981564797, −4.553390695242179, −4.006969365911414, −3.408940009250916, −2.751380974540420, −2.341659403335786, −2.000415955517083, −0.9931914633863256, −0.4583056883235691,
0.4583056883235691, 0.9931914633863256, 2.000415955517083, 2.341659403335786, 2.751380974540420, 3.408940009250916, 4.006969365911414, 4.553390695242179, 4.677559981564797, 5.718329572009714, 5.989172012548128, 6.424150791019623, 7.104783899826098, 7.530709056425783, 7.866163469723923, 8.453190966304024, 8.837781422743307, 9.355946871656299, 9.821412838703098, 10.08337044349851, 10.76051664150381, 10.97415553724217, 11.72801877163942, 12.12272969205445, 12.50866831685527
