| L(s) = 1 | + 1.66·3-s − 3.27·5-s − 4.00·7-s − 0.217·9-s − 2.76·11-s + 1.91·13-s − 5.45·15-s − 7.42·17-s − 0.351·19-s − 6.67·21-s + 6.03·23-s + 5.69·25-s − 5.36·27-s − 4.38·29-s − 6.97·31-s − 4.60·33-s + 13.0·35-s + 9.17·37-s + 3.19·39-s + 10.3·41-s + 3.16·43-s + 0.711·45-s + 7.23·47-s + 9.01·49-s − 12.3·51-s + 3.87·53-s + 9.03·55-s + ⋯ |
| L(s) = 1 | + 0.963·3-s − 1.46·5-s − 1.51·7-s − 0.0725·9-s − 0.833·11-s + 0.530·13-s − 1.40·15-s − 1.80·17-s − 0.0806·19-s − 1.45·21-s + 1.25·23-s + 1.13·25-s − 1.03·27-s − 0.813·29-s − 1.25·31-s − 0.802·33-s + 2.21·35-s + 1.50·37-s + 0.511·39-s + 1.61·41-s + 0.482·43-s + 0.106·45-s + 1.05·47-s + 1.28·49-s − 1.73·51-s + 0.532·53-s + 1.21·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8214496620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8214496620\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 1.66T + 3T^{2} \) |
| 5 | \( 1 + 3.27T + 5T^{2} \) |
| 7 | \( 1 + 4.00T + 7T^{2} \) |
| 11 | \( 1 + 2.76T + 11T^{2} \) |
| 13 | \( 1 - 1.91T + 13T^{2} \) |
| 17 | \( 1 + 7.42T + 17T^{2} \) |
| 19 | \( 1 + 0.351T + 19T^{2} \) |
| 23 | \( 1 - 6.03T + 23T^{2} \) |
| 29 | \( 1 + 4.38T + 29T^{2} \) |
| 31 | \( 1 + 6.97T + 31T^{2} \) |
| 37 | \( 1 - 9.17T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 3.16T + 43T^{2} \) |
| 47 | \( 1 - 7.23T + 47T^{2} \) |
| 53 | \( 1 - 3.87T + 53T^{2} \) |
| 59 | \( 1 - 5.47T + 59T^{2} \) |
| 61 | \( 1 - 4.49T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 - 9.81T + 79T^{2} \) |
| 83 | \( 1 - 6.01T + 83T^{2} \) |
| 89 | \( 1 - 3.36T + 89T^{2} \) |
| 97 | \( 1 + 7.37T + 97T^{2} \) |
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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
−8.528651302582131953605652510301, −7.64579198257746735870973604848, −7.25753271699147056843776739576, −6.39031607167877120984332404623, −5.51502378271603517188241744779, −4.22514941567769348930038174998, −3.82429698823647741163377792715, −2.94669546821759399726244782751, −2.43797015282357504736071119870, −0.45891607561949084402640184236,
0.45891607561949084402640184236, 2.43797015282357504736071119870, 2.94669546821759399726244782751, 3.82429698823647741163377792715, 4.22514941567769348930038174998, 5.51502378271603517188241744779, 6.39031607167877120984332404623, 7.25753271699147056843776739576, 7.64579198257746735870973604848, 8.528651302582131953605652510301
