| L(s) = 1 | − 5·5-s + 34·7-s − 48·11-s − 70·13-s + 27·17-s − 119·19-s − 51·23-s + 25·25-s + 30·29-s + 133·31-s − 170·35-s + 218·37-s − 156·41-s + 88·43-s − 516·47-s + 813·49-s + 639·53-s + 240·55-s − 654·59-s + 461·61-s + 350·65-s − 182·67-s + 900·71-s + 704·73-s − 1.63e3·77-s + 1.37e3·79-s + 915·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.83·7-s − 1.31·11-s − 1.49·13-s + 0.385·17-s − 1.43·19-s − 0.462·23-s + 1/5·25-s + 0.192·29-s + 0.770·31-s − 0.821·35-s + 0.968·37-s − 0.594·41-s + 0.312·43-s − 1.60·47-s + 2.37·49-s + 1.65·53-s + 0.588·55-s − 1.44·59-s + 0.967·61-s + 0.667·65-s − 0.331·67-s + 1.50·71-s + 1.12·73-s − 2.41·77-s + 1.95·79-s + 1.21·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.738103272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.738103272\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 - 34 T + p^{3} T^{2} \) |
| 11 | \( 1 + 48 T + p^{3} T^{2} \) |
| 13 | \( 1 + 70 T + p^{3} T^{2} \) |
| 17 | \( 1 - 27 T + p^{3} T^{2} \) |
| 19 | \( 1 + 119 T + p^{3} T^{2} \) |
| 23 | \( 1 + 51 T + p^{3} T^{2} \) |
| 29 | \( 1 - 30 T + p^{3} T^{2} \) |
| 31 | \( 1 - 133 T + p^{3} T^{2} \) |
| 37 | \( 1 - 218 T + p^{3} T^{2} \) |
| 41 | \( 1 + 156 T + p^{3} T^{2} \) |
| 43 | \( 1 - 88 T + p^{3} T^{2} \) |
| 47 | \( 1 + 516 T + p^{3} T^{2} \) |
| 53 | \( 1 - 639 T + p^{3} T^{2} \) |
| 59 | \( 1 + 654 T + p^{3} T^{2} \) |
| 61 | \( 1 - 461 T + p^{3} T^{2} \) |
| 67 | \( 1 + 182 T + p^{3} T^{2} \) |
| 71 | \( 1 - 900 T + p^{3} T^{2} \) |
| 73 | \( 1 - 704 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1375 T + p^{3} T^{2} \) |
| 83 | \( 1 - 915 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1116 T + p^{3} T^{2} \) |
| 97 | \( 1 + 16 T + p^{3} T^{2} \) |
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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
−8.314628810769755394264317742322, −8.030521712637963247686870902223, −7.48452784400259632114601247994, −6.42011502422002326179056406067, −5.10749664892860343742006014951, −4.94559678893672419995521303350, −4.02290537273686512713053240469, −2.58059826102249581274302375256, −1.99464082487382479317995178489, −0.57933029289967145268415446071,
0.57933029289967145268415446071, 1.99464082487382479317995178489, 2.58059826102249581274302375256, 4.02290537273686512713053240469, 4.94559678893672419995521303350, 5.10749664892860343742006014951, 6.42011502422002326179056406067, 7.48452784400259632114601247994, 8.030521712637963247686870902223, 8.314628810769755394264317742322
