| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s − 13-s + 15-s − 17-s − 4·19-s − 21-s + 4·23-s + 25-s + 27-s − 2·29-s + 8·31-s − 35-s − 6·37-s − 39-s + 2·41-s − 8·43-s + 45-s + 4·47-s + 49-s − 51-s + 6·53-s − 4·57-s + 4·59-s − 6·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s − 0.242·17-s − 0.917·19-s − 0.218·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.169·35-s − 0.986·37-s − 0.160·39-s + 0.312·41-s − 1.21·43-s + 0.149·45-s + 0.583·47-s + 1/7·49-s − 0.140·51-s + 0.824·53-s − 0.529·57-s + 0.520·59-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 371280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 371280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.308090893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.308090893\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−12.62124896429770, −12.06872743787515, −11.66993397775854, −11.09533766731988, −10.52976712621495, −10.20823468403879, −9.843740158120706, −9.205965436170169, −8.955601385037068, −8.337802680852057, −8.168164259342455, −7.328626129772932, −6.891371037945438, −6.700427509385296, −5.949057704055330, −5.588850738803568, −4.906909893761798, −4.481039266722721, −3.948914157271222, −3.377459397459965, −2.699031889965851, −2.528415112029274, −1.721701136831960, −1.247761177040416, −0.3673486934105001,
0.3673486934105001, 1.247761177040416, 1.721701136831960, 2.528415112029274, 2.699031889965851, 3.377459397459965, 3.948914157271222, 4.481039266722721, 4.906909893761798, 5.588850738803568, 5.949057704055330, 6.700427509385296, 6.891371037945438, 7.328626129772932, 8.168164259342455, 8.337802680852057, 8.955601385037068, 9.205965436170169, 9.843740158120706, 10.20823468403879, 10.52976712621495, 11.09533766731988, 11.66993397775854, 12.06872743787515, 12.62124896429770
