| L(s) = 1 | + 3.06·3-s + 3.33·5-s − 1.87·7-s + 6.41·9-s − 4.57·11-s + 6.66·13-s + 10.2·15-s + 6.03·17-s + 19-s − 5.75·21-s − 5.94·23-s + 6.09·25-s + 10.4·27-s − 4.37·29-s + 4.07·31-s − 14.0·33-s − 6.24·35-s + 7.28·37-s + 20.4·39-s + 5.42·41-s − 1.63·43-s + 21.3·45-s − 11.0·47-s − 3.48·49-s + 18.5·51-s + 0.851·53-s − 15.2·55-s + ⋯ |
| L(s) = 1 | + 1.77·3-s + 1.48·5-s − 0.708·7-s + 2.13·9-s − 1.38·11-s + 1.84·13-s + 2.63·15-s + 1.46·17-s + 0.229·19-s − 1.25·21-s − 1.23·23-s + 1.21·25-s + 2.01·27-s − 0.813·29-s + 0.732·31-s − 2.44·33-s − 1.05·35-s + 1.19·37-s + 3.27·39-s + 0.847·41-s − 0.248·43-s + 3.18·45-s − 1.61·47-s − 0.497·49-s + 2.59·51-s + 0.116·53-s − 2.05·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.318418554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.318418554\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 3 | \( 1 - 3.06T + 3T^{2} \) |
| 5 | \( 1 - 3.33T + 5T^{2} \) |
| 7 | \( 1 + 1.87T + 7T^{2} \) |
| 11 | \( 1 + 4.57T + 11T^{2} \) |
| 13 | \( 1 - 6.66T + 13T^{2} \) |
| 17 | \( 1 - 6.03T + 17T^{2} \) |
| 23 | \( 1 + 5.94T + 23T^{2} \) |
| 29 | \( 1 + 4.37T + 29T^{2} \) |
| 31 | \( 1 - 4.07T + 31T^{2} \) |
| 37 | \( 1 - 7.28T + 37T^{2} \) |
| 41 | \( 1 - 5.42T + 41T^{2} \) |
| 43 | \( 1 + 1.63T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 - 0.851T + 53T^{2} \) |
| 59 | \( 1 - 8.17T + 59T^{2} \) |
| 61 | \( 1 - 0.392T + 61T^{2} \) |
| 67 | \( 1 + 8.78T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 + 3.83T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−8.032554467879621760483532578203, −7.77495785193050160971165989815, −6.61572938504970711674296525414, −5.96117212560839007880253988911, −5.39888998664277793599768682853, −4.14925425277144652153773778589, −3.32823719045493820303263156317, −2.84299962620681327632014741130, −2.02567030677653399215904152363, −1.23345535642388836753281294469,
1.23345535642388836753281294469, 2.02567030677653399215904152363, 2.84299962620681327632014741130, 3.32823719045493820303263156317, 4.14925425277144652153773778589, 5.39888998664277793599768682853, 5.96117212560839007880253988911, 6.61572938504970711674296525414, 7.77495785193050160971165989815, 8.032554467879621760483532578203
