| L(s) = 1 | + 3-s − 2·5-s − 7-s + 9-s + 2·13-s − 2·15-s + 17-s − 21-s − 25-s + 27-s + 2·29-s + 8·31-s + 2·35-s − 6·37-s + 2·39-s − 6·41-s + 4·43-s − 2·45-s + 49-s + 51-s + 14·53-s − 8·59-s + 14·61-s − 63-s − 4·65-s + 4·67-s + 8·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.554·13-s − 0.516·15-s + 0.242·17-s − 0.218·21-s − 1/5·25-s + 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.338·35-s − 0.986·37-s + 0.320·39-s − 0.937·41-s + 0.609·43-s − 0.298·45-s + 1/7·49-s + 0.140·51-s + 1.92·53-s − 1.04·59-s + 1.79·61-s − 0.125·63-s − 0.496·65-s + 0.488·67-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.798268217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.798268217\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.455731945072633336414656016150, −8.293523860811542037254609471509, −7.29504998230991219563900037857, −6.70141229159940842717593963790, −5.74399564630477680169234153812, −4.71126394239504219168220825871, −3.84431934253247134888356945886, −3.28615585654671725260821047345, −2.21359341558133614477796467457, −0.806546035797645823668139895748,
0.806546035797645823668139895748, 2.21359341558133614477796467457, 3.28615585654671725260821047345, 3.84431934253247134888356945886, 4.71126394239504219168220825871, 5.74399564630477680169234153812, 6.70141229159940842717593963790, 7.29504998230991219563900037857, 8.293523860811542037254609471509, 8.455731945072633336414656016150
