| L(s) = 1 | + 3.35·3-s − 3.67·5-s − 1.57·7-s + 8.24·9-s − 0.611·11-s − 13-s − 12.3·15-s − 0.239·17-s + 1.75·19-s − 5.26·21-s − 5.67·23-s + 8.52·25-s + 17.5·27-s + 29-s + 9.32·31-s − 2.04·33-s + 5.78·35-s − 2.03·37-s − 3.35·39-s + 0.649·41-s + 10.3·43-s − 30.3·45-s + 6.73·47-s − 4.52·49-s − 0.803·51-s − 0.0167·53-s + 2.24·55-s + ⋯ |
| L(s) = 1 | + 1.93·3-s − 1.64·5-s − 0.594·7-s + 2.74·9-s − 0.184·11-s − 0.277·13-s − 3.18·15-s − 0.0581·17-s + 0.402·19-s − 1.14·21-s − 1.18·23-s + 1.70·25-s + 3.38·27-s + 0.185·29-s + 1.67·31-s − 0.356·33-s + 0.977·35-s − 0.334·37-s − 0.536·39-s + 0.101·41-s + 1.57·43-s − 4.51·45-s + 0.981·47-s − 0.647·49-s − 0.112·51-s − 0.00230·53-s + 0.303·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.798313476\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.798313476\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 3.35T + 3T^{2} \) |
| 5 | \( 1 + 3.67T + 5T^{2} \) |
| 7 | \( 1 + 1.57T + 7T^{2} \) |
| 11 | \( 1 + 0.611T + 11T^{2} \) |
| 17 | \( 1 + 0.239T + 17T^{2} \) |
| 19 | \( 1 - 1.75T + 19T^{2} \) |
| 23 | \( 1 + 5.67T + 23T^{2} \) |
| 31 | \( 1 - 9.32T + 31T^{2} \) |
| 37 | \( 1 + 2.03T + 37T^{2} \) |
| 41 | \( 1 - 0.649T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 - 6.73T + 47T^{2} \) |
| 53 | \( 1 + 0.0167T + 53T^{2} \) |
| 59 | \( 1 - 2.88T + 59T^{2} \) |
| 61 | \( 1 - 9.20T + 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 7.85T + 73T^{2} \) |
| 79 | \( 1 - 15.2T + 79T^{2} \) |
| 83 | \( 1 - 9.20T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 4.79T + 97T^{2} \) |
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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
−7.904738245612868998843837164377, −7.69822096087573552806844642948, −7.07580689069440792183596091516, −6.19449810914752138349881249688, −4.69701812619846221894967659826, −4.19960357182500187630993784150, −3.51732148604443634579065445831, −2.97116390740543949084166155809, −2.18906957593139910873001017666, −0.790893099276880608581936049301,
0.790893099276880608581936049301, 2.18906957593139910873001017666, 2.97116390740543949084166155809, 3.51732148604443634579065445831, 4.19960357182500187630993784150, 4.69701812619846221894967659826, 6.19449810914752138349881249688, 7.07580689069440792183596091516, 7.69822096087573552806844642948, 7.904738245612868998843837164377
