| L(s) = 1 | − 2.93·3-s − 5-s + 4.42·7-s + 5.60·9-s − 4.88·11-s + 5.14·13-s + 2.93·15-s + 5.88·17-s + 1.35·19-s − 12.9·21-s + 6.77·23-s + 25-s − 7.64·27-s + 10.1·29-s − 7.69·31-s + 14.3·33-s − 4.42·35-s + 10.4·37-s − 15.0·39-s + 1.04·41-s + 1.57·43-s − 5.60·45-s + 10.9·47-s + 12.6·49-s − 17.2·51-s + 8.65·53-s + 4.88·55-s + ⋯ |
| L(s) = 1 | − 1.69·3-s − 0.447·5-s + 1.67·7-s + 1.86·9-s − 1.47·11-s + 1.42·13-s + 0.757·15-s + 1.42·17-s + 0.310·19-s − 2.83·21-s + 1.41·23-s + 0.200·25-s − 1.47·27-s + 1.89·29-s − 1.38·31-s + 2.49·33-s − 0.748·35-s + 1.71·37-s − 2.41·39-s + 0.163·41-s + 0.240·43-s − 0.835·45-s + 1.59·47-s + 1.80·49-s − 2.41·51-s + 1.18·53-s + 0.658·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.595864827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.595864827\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 + 2.93T + 3T^{2} \) |
| 7 | \( 1 - 4.42T + 7T^{2} \) |
| 11 | \( 1 + 4.88T + 11T^{2} \) |
| 13 | \( 1 - 5.14T + 13T^{2} \) |
| 17 | \( 1 - 5.88T + 17T^{2} \) |
| 19 | \( 1 - 1.35T + 19T^{2} \) |
| 23 | \( 1 - 6.77T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 + 7.69T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 1.04T + 41T^{2} \) |
| 43 | \( 1 - 1.57T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 - 8.65T + 53T^{2} \) |
| 59 | \( 1 - 0.162T + 59T^{2} \) |
| 61 | \( 1 + 8.71T + 61T^{2} \) |
| 67 | \( 1 + 9.19T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 + 7.41T + 73T^{2} \) |
| 79 | \( 1 - 0.446T + 79T^{2} \) |
| 83 | \( 1 + 2.85T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 - 2.63T + 97T^{2} \) |
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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
−7.56621902711694747027767874181, −7.38576126669845916570928323950, −6.17133835648980857151148693218, −5.67426234123063808103825800473, −5.03920412812011096559207434514, −4.69089989698945215802996180069, −3.74415840635498513509179288558, −2.61547403250891314030028110418, −1.20550163576508371003020874839, −0.865543162703537541910894925099,
0.865543162703537541910894925099, 1.20550163576508371003020874839, 2.61547403250891314030028110418, 3.74415840635498513509179288558, 4.69089989698945215802996180069, 5.03920412812011096559207434514, 5.67426234123063808103825800473, 6.17133835648980857151148693218, 7.38576126669845916570928323950, 7.56621902711694747027767874181
