| L(s) = 1 | + 3-s − 3.44·5-s + 4.67·7-s + 9-s − 3.77·11-s − 3.51·13-s − 3.44·15-s − 8.12·17-s + 5.33·19-s + 4.67·21-s − 23-s + 6.87·25-s + 27-s + 29-s − 9.72·31-s − 3.77·33-s − 16.1·35-s + 7.45·37-s − 3.51·39-s + 3.95·41-s − 2.47·43-s − 3.44·45-s − 3.53·47-s + 14.8·49-s − 8.12·51-s + 9.38·53-s + 13.0·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.54·5-s + 1.76·7-s + 0.333·9-s − 1.13·11-s − 0.973·13-s − 0.889·15-s − 1.97·17-s + 1.22·19-s + 1.02·21-s − 0.208·23-s + 1.37·25-s + 0.192·27-s + 0.185·29-s − 1.74·31-s − 0.657·33-s − 2.72·35-s + 1.22·37-s − 0.562·39-s + 0.617·41-s − 0.377·43-s − 0.513·45-s − 0.515·47-s + 2.12·49-s − 1.13·51-s + 1.28·53-s + 1.75·55-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
| \(L(1)\) |
\(\approx\) |
\(1.612667168\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.612667168\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.44T + 5T^{2} \) |
| 7 | \( 1 - 4.67T + 7T^{2} \) |
| 11 | \( 1 + 3.77T + 11T^{2} \) |
| 13 | \( 1 + 3.51T + 13T^{2} \) |
| 17 | \( 1 + 8.12T + 17T^{2} \) |
| 19 | \( 1 - 5.33T + 19T^{2} \) |
| 31 | \( 1 + 9.72T + 31T^{2} \) |
| 37 | \( 1 - 7.45T + 37T^{2} \) |
| 41 | \( 1 - 3.95T + 41T^{2} \) |
| 43 | \( 1 + 2.47T + 43T^{2} \) |
| 47 | \( 1 + 3.53T + 47T^{2} \) |
| 53 | \( 1 - 9.38T + 53T^{2} \) |
| 59 | \( 1 - 0.971T + 59T^{2} \) |
| 61 | \( 1 + 12.8T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 - 9.25T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−7.81794115829061540492752964967, −7.48568565513120486433571331888, −6.83429499185307302313173523012, −5.40577414362700328945690285886, −4.86336765764895200775806500173, −4.36264586450237781327951255794, −3.62476943573702264527268127206, −2.55881269106374937417775387703, −1.98756351175553465784189286997, −0.59173008312922835169941363516,
0.59173008312922835169941363516, 1.98756351175553465784189286997, 2.55881269106374937417775387703, 3.62476943573702264527268127206, 4.36264586450237781327951255794, 4.86336765764895200775806500173, 5.40577414362700328945690285886, 6.83429499185307302313173523012, 7.48568565513120486433571331888, 7.81794115829061540492752964967
