| L(s) = 1 | + 1.39·2-s − 0.0249·3-s − 0.0413·4-s + 1.82·5-s − 0.0349·6-s − 3.45·7-s − 2.85·8-s − 2.99·9-s + 2.54·10-s + 0.872·11-s + 0.00103·12-s − 1.44·13-s − 4.83·14-s − 0.0455·15-s − 3.91·16-s − 4.03·17-s − 4.19·18-s − 19-s − 0.0753·20-s + 0.0863·21-s + 1.22·22-s + 5.71·23-s + 0.0714·24-s − 1.68·25-s − 2.01·26-s + 0.149·27-s + 0.142·28-s + ⋯ |
| L(s) = 1 | + 0.989·2-s − 0.0144·3-s − 0.0206·4-s + 0.814·5-s − 0.0142·6-s − 1.30·7-s − 1.01·8-s − 0.999·9-s + 0.806·10-s + 0.262·11-s + 0.000298·12-s − 0.400·13-s − 1.29·14-s − 0.0117·15-s − 0.978·16-s − 0.977·17-s − 0.989·18-s − 0.229·19-s − 0.0168·20-s + 0.0188·21-s + 0.260·22-s + 1.19·23-s + 0.0145·24-s − 0.336·25-s − 0.396·26-s + 0.0288·27-s + 0.0269·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.936836723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.936836723\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + T \) |
| 317 | \( 1 - T \) |
| good | 2 | \( 1 - 1.39T + 2T^{2} \) |
| 3 | \( 1 + 0.0249T + 3T^{2} \) |
| 5 | \( 1 - 1.82T + 5T^{2} \) |
| 7 | \( 1 + 3.45T + 7T^{2} \) |
| 11 | \( 1 - 0.872T + 11T^{2} \) |
| 13 | \( 1 + 1.44T + 13T^{2} \) |
| 17 | \( 1 + 4.03T + 17T^{2} \) |
| 23 | \( 1 - 5.71T + 23T^{2} \) |
| 29 | \( 1 - 4.29T + 29T^{2} \) |
| 31 | \( 1 - 6.19T + 31T^{2} \) |
| 37 | \( 1 - 2.36T + 37T^{2} \) |
| 41 | \( 1 - 1.07T + 41T^{2} \) |
| 43 | \( 1 - 5.95T + 43T^{2} \) |
| 47 | \( 1 - 0.423T + 47T^{2} \) |
| 53 | \( 1 + 7.54T + 53T^{2} \) |
| 59 | \( 1 - 0.740T + 59T^{2} \) |
| 61 | \( 1 - 8.93T + 61T^{2} \) |
| 67 | \( 1 - 2.65T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 - 2.54T + 97T^{2} \) |
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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
−8.208006504741592149305891743512, −6.94580821801393767572904940392, −6.34873584103002561727347364598, −6.01928238697567969835993730180, −5.19243443882936215020216884872, −4.53361526434431906445992660533, −3.61513248468517939241975037379, −2.84257229906783158379018535878, −2.37234334506413757973349913982, −0.59455486214015940247421368760,
0.59455486214015940247421368760, 2.37234334506413757973349913982, 2.84257229906783158379018535878, 3.61513248468517939241975037379, 4.53361526434431906445992660533, 5.19243443882936215020216884872, 6.01928238697567969835993730180, 6.34873584103002561727347364598, 6.94580821801393767572904940392, 8.208006504741592149305891743512
