| L(s) = 1 | + 0.792·3-s − 3.46·7-s − 2.37·9-s + 11-s + 1.58·17-s − 4·19-s − 2.74·21-s − 0.792·23-s − 4.25·27-s + 8.74·29-s − 3.37·31-s + 0.792·33-s − 1.08·37-s + 8.74·41-s − 3.46·43-s + 6.63·47-s + 4.99·49-s + 1.25·51-s + 10.0·53-s − 3.16·57-s + 7.37·59-s − 0.744·61-s + 8.21·63-s − 9.30·67-s − 0.627·69-s + 10.1·71-s − 6.92·73-s + ⋯ |
| L(s) = 1 | + 0.457·3-s − 1.30·7-s − 0.790·9-s + 0.301·11-s + 0.384·17-s − 0.917·19-s − 0.598·21-s − 0.165·23-s − 0.819·27-s + 1.62·29-s − 0.605·31-s + 0.137·33-s − 0.178·37-s + 1.36·41-s − 0.528·43-s + 0.967·47-s + 0.714·49-s + 0.175·51-s + 1.38·53-s − 0.419·57-s + 0.959·59-s − 0.0953·61-s + 1.03·63-s − 1.13·67-s − 0.0755·69-s + 1.20·71-s − 0.810·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.488140999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.488140999\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 0.792T + 3T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 1.58T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 0.792T + 23T^{2} \) |
| 29 | \( 1 - 8.74T + 29T^{2} \) |
| 31 | \( 1 + 3.37T + 31T^{2} \) |
| 37 | \( 1 + 1.08T + 37T^{2} \) |
| 41 | \( 1 - 8.74T + 41T^{2} \) |
| 43 | \( 1 + 3.46T + 43T^{2} \) |
| 47 | \( 1 - 6.63T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 7.37T + 59T^{2} \) |
| 61 | \( 1 + 0.744T + 61T^{2} \) |
| 67 | \( 1 + 9.30T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 + 1.25T + 79T^{2} \) |
| 83 | \( 1 + 6.63T + 83T^{2} \) |
| 89 | \( 1 - 1.37T + 89T^{2} \) |
| 97 | \( 1 - 5.84T + 97T^{2} \) |
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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
−8.625532270970599991352435804858, −7.63832764812210769478900359322, −6.87037408836958828399530944787, −6.15785418336555533605244835119, −5.63596220592554121554653069722, −4.46263131132669701585907536196, −3.63852695119969555239119150797, −2.94842817339672607497200936411, −2.19553695945643343039731327788, −0.64111603898426997818886381502,
0.64111603898426997818886381502, 2.19553695945643343039731327788, 2.94842817339672607497200936411, 3.63852695119969555239119150797, 4.46263131132669701585907536196, 5.63596220592554121554653069722, 6.15785418336555533605244835119, 6.87037408836958828399530944787, 7.63832764812210769478900359322, 8.625532270970599991352435804858
