| L(s) = 1 | − 1.30·3-s − 0.302·5-s + 7-s − 1.30·9-s + 4.60·11-s + 4·13-s + 0.394·15-s + 17-s + 4.60·19-s − 1.30·21-s + 6·23-s − 4.90·25-s + 5.60·27-s + 8.60·29-s + 4.90·31-s − 6·33-s − 0.302·35-s + 4.60·37-s − 5.21·39-s + 0.697·41-s − 2.69·43-s + 0.394·45-s − 9.21·47-s + 49-s − 1.30·51-s + 7.69·53-s − 1.39·55-s + ⋯ |
| L(s) = 1 | − 0.752·3-s − 0.135·5-s + 0.377·7-s − 0.434·9-s + 1.38·11-s + 1.10·13-s + 0.101·15-s + 0.242·17-s + 1.05·19-s − 0.284·21-s + 1.25·23-s − 0.981·25-s + 1.07·27-s + 1.59·29-s + 0.881·31-s − 1.04·33-s − 0.0511·35-s + 0.757·37-s − 0.834·39-s + 0.108·41-s − 0.411·43-s + 0.0588·45-s − 1.34·47-s + 0.142·49-s − 0.182·51-s + 1.05·53-s − 0.188·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.013290245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.013290245\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 1.30T + 3T^{2} \) |
| 5 | \( 1 + 0.302T + 5T^{2} \) |
| 11 | \( 1 - 4.60T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 19 | \( 1 - 4.60T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 8.60T + 29T^{2} \) |
| 31 | \( 1 - 4.90T + 31T^{2} \) |
| 37 | \( 1 - 4.60T + 37T^{2} \) |
| 41 | \( 1 - 0.697T + 41T^{2} \) |
| 43 | \( 1 + 2.69T + 43T^{2} \) |
| 47 | \( 1 + 9.21T + 47T^{2} \) |
| 53 | \( 1 - 7.69T + 53T^{2} \) |
| 59 | \( 1 + 5.21T + 59T^{2} \) |
| 61 | \( 1 - 4.69T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 + 0.605T + 71T^{2} \) |
| 73 | \( 1 + 5.90T + 73T^{2} \) |
| 79 | \( 1 - 1.21T + 79T^{2} \) |
| 83 | \( 1 + 17.8T + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 - 4.30T + 97T^{2} \) |
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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
−7.895558826599912892279585815090, −7.04577750145546687730663919234, −6.30661371649266310320742525485, −5.97224574128897542816915031891, −5.05190671998961460900172891739, −4.45180787246456566322049477461, −3.53617164722893198025667669457, −2.84722384955827859708488149548, −1.40385363344112298267582649701, −0.852598322904651585987461620056,
0.852598322904651585987461620056, 1.40385363344112298267582649701, 2.84722384955827859708488149548, 3.53617164722893198025667669457, 4.45180787246456566322049477461, 5.05190671998961460900172891739, 5.97224574128897542816915031891, 6.30661371649266310320742525485, 7.04577750145546687730663919234, 7.895558826599912892279585815090
