| L(s) = 1 | − 0.828·3-s + 0.378·5-s − 3.91·7-s − 2.31·9-s + 4.10·11-s − 0.0890·13-s − 0.313·15-s + 2.16·17-s − 19-s + 3.24·21-s − 3.36·23-s − 4.85·25-s + 4.40·27-s + 0.979·29-s + 0.866·31-s − 3.39·33-s − 1.48·35-s + 2.32·37-s + 0.0737·39-s − 8.03·41-s − 4.18·43-s − 0.874·45-s + 4.66·47-s + 8.33·49-s − 1.79·51-s − 53-s + 1.55·55-s + ⋯ |
| L(s) = 1 | − 0.478·3-s + 0.169·5-s − 1.47·7-s − 0.771·9-s + 1.23·11-s − 0.0247·13-s − 0.0808·15-s + 0.525·17-s − 0.229·19-s + 0.707·21-s − 0.701·23-s − 0.971·25-s + 0.847·27-s + 0.181·29-s + 0.155·31-s − 0.591·33-s − 0.250·35-s + 0.381·37-s + 0.0118·39-s − 1.25·41-s − 0.638·43-s − 0.130·45-s + 0.680·47-s + 1.19·49-s − 0.251·51-s − 0.137·53-s + 0.209·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9839359160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9839359160\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| good | 3 | \( 1 + 0.828T + 3T^{2} \) |
| 5 | \( 1 - 0.378T + 5T^{2} \) |
| 7 | \( 1 + 3.91T + 7T^{2} \) |
| 11 | \( 1 - 4.10T + 11T^{2} \) |
| 13 | \( 1 + 0.0890T + 13T^{2} \) |
| 17 | \( 1 - 2.16T + 17T^{2} \) |
| 23 | \( 1 + 3.36T + 23T^{2} \) |
| 29 | \( 1 - 0.979T + 29T^{2} \) |
| 31 | \( 1 - 0.866T + 31T^{2} \) |
| 37 | \( 1 - 2.32T + 37T^{2} \) |
| 41 | \( 1 + 8.03T + 41T^{2} \) |
| 43 | \( 1 + 4.18T + 43T^{2} \) |
| 47 | \( 1 - 4.66T + 47T^{2} \) |
| 59 | \( 1 + 0.0391T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 2.22T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 + 5.23T + 73T^{2} \) |
| 79 | \( 1 + 4.14T + 79T^{2} \) |
| 83 | \( 1 - 6.30T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 - 2.09T + 97T^{2} \) |
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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
−8.586672700007616408266136201985, −7.64476475796727651408154557325, −6.70183951570641981110684406270, −6.19390038290849232375517199177, −5.78848406767157520595484458454, −4.69662083243057744558255134691, −3.67313560615961912697850657374, −3.15055717658675109379284054095, −1.95630313677855422583490693850, −0.56346023551132534980163144097,
0.56346023551132534980163144097, 1.95630313677855422583490693850, 3.15055717658675109379284054095, 3.67313560615961912697850657374, 4.69662083243057744558255134691, 5.78848406767157520595484458454, 6.19390038290849232375517199177, 6.70183951570641981110684406270, 7.64476475796727651408154557325, 8.586672700007616408266136201985
