| L(s) = 1 | − 3-s − 1.42·5-s − 0.237·7-s + 9-s − 4.47·11-s + 2.57·13-s + 1.42·15-s + 5.67·17-s + 3.71·19-s + 0.237·21-s − 23-s − 2.96·25-s − 27-s − 29-s − 8.28·31-s + 4.47·33-s + 0.339·35-s + 9.51·37-s − 2.57·39-s + 6.95·41-s − 9.33·43-s − 1.42·45-s + 7.70·47-s − 6.94·49-s − 5.67·51-s + 0.152·53-s + 6.39·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.638·5-s − 0.0899·7-s + 0.333·9-s − 1.35·11-s + 0.714·13-s + 0.368·15-s + 1.37·17-s + 0.851·19-s + 0.0519·21-s − 0.208·23-s − 0.592·25-s − 0.192·27-s − 0.185·29-s − 1.48·31-s + 0.779·33-s + 0.0573·35-s + 1.56·37-s − 0.412·39-s + 1.08·41-s − 1.42·43-s − 0.212·45-s + 1.12·47-s − 0.991·49-s − 0.795·51-s + 0.0209·53-s + 0.861·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.067186374\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.067186374\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 + 1.42T + 5T^{2} \) |
| 7 | \( 1 + 0.237T + 7T^{2} \) |
| 11 | \( 1 + 4.47T + 11T^{2} \) |
| 13 | \( 1 - 2.57T + 13T^{2} \) |
| 17 | \( 1 - 5.67T + 17T^{2} \) |
| 19 | \( 1 - 3.71T + 19T^{2} \) |
| 31 | \( 1 + 8.28T + 31T^{2} \) |
| 37 | \( 1 - 9.51T + 37T^{2} \) |
| 41 | \( 1 - 6.95T + 41T^{2} \) |
| 43 | \( 1 + 9.33T + 43T^{2} \) |
| 47 | \( 1 - 7.70T + 47T^{2} \) |
| 53 | \( 1 - 0.152T + 53T^{2} \) |
| 59 | \( 1 + 4.39T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 - 15.9T + 83T^{2} \) |
| 89 | \( 1 - 2.08T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−7.73809150362909994354023409609, −7.39653725547080706711779252956, −6.33628002357970020817225722459, −5.61667626300639756095639306789, −5.25672164295986058790612522761, −4.26767414859852784504974433793, −3.53905437101032146653671132783, −2.82954499170641076111942818356, −1.60764212553167372785051260860, −0.53730830027196068276691866973,
0.53730830027196068276691866973, 1.60764212553167372785051260860, 2.82954499170641076111942818356, 3.53905437101032146653671132783, 4.26767414859852784504974433793, 5.25672164295986058790612522761, 5.61667626300639756095639306789, 6.33628002357970020817225722459, 7.39653725547080706711779252956, 7.73809150362909994354023409609
