| L(s) = 1 | + 1.56·3-s + 3.44·5-s + 1.77·7-s − 0.565·9-s + 0.674·11-s − 2.56·13-s + 5.37·15-s + 7.81·17-s + 3.57·19-s + 2.76·21-s + 4.75·23-s + 6.87·25-s − 5.56·27-s − 8.77·29-s + 4.12·31-s + 1.05·33-s + 6.10·35-s − 8.08·37-s − 4.00·39-s + 2.30·41-s − 3.23·43-s − 1.94·45-s + 10.7·47-s − 3.85·49-s + 12.1·51-s + 11.7·53-s + 2.32·55-s + ⋯ |
| L(s) = 1 | + 0.900·3-s + 1.54·5-s + 0.670·7-s − 0.188·9-s + 0.203·11-s − 0.711·13-s + 1.38·15-s + 1.89·17-s + 0.820·19-s + 0.603·21-s + 0.992·23-s + 1.37·25-s − 1.07·27-s − 1.62·29-s + 0.740·31-s + 0.183·33-s + 1.03·35-s − 1.32·37-s − 0.640·39-s + 0.360·41-s − 0.493·43-s − 0.290·45-s + 1.57·47-s − 0.550·49-s + 1.70·51-s + 1.61·53-s + 0.313·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.022276126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.022276126\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 5 | \( 1 - 3.44T + 5T^{2} \) |
| 7 | \( 1 - 1.77T + 7T^{2} \) |
| 11 | \( 1 - 0.674T + 11T^{2} \) |
| 13 | \( 1 + 2.56T + 13T^{2} \) |
| 17 | \( 1 - 7.81T + 17T^{2} \) |
| 19 | \( 1 - 3.57T + 19T^{2} \) |
| 23 | \( 1 - 4.75T + 23T^{2} \) |
| 29 | \( 1 + 8.77T + 29T^{2} \) |
| 31 | \( 1 - 4.12T + 31T^{2} \) |
| 37 | \( 1 + 8.08T + 37T^{2} \) |
| 41 | \( 1 - 2.30T + 41T^{2} \) |
| 43 | \( 1 + 3.23T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 - 6.36T + 67T^{2} \) |
| 71 | \( 1 + 5.50T + 71T^{2} \) |
| 73 | \( 1 - 0.424T + 73T^{2} \) |
| 79 | \( 1 - 3.50T + 79T^{2} \) |
| 83 | \( 1 + 1.14T + 83T^{2} \) |
| 89 | \( 1 - 9.02T + 89T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
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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
−8.596625433208947405879267301543, −7.62588663573730151592930986823, −7.28018757441796852813299079351, −6.01636785271938578895132479284, −5.48456408092783435616763838861, −4.91600442197387406585931308504, −3.55092888160363787798225954509, −2.87277240523447263591486231079, −2.01443083663578714303722357226, −1.23097189048005913058699570280,
1.23097189048005913058699570280, 2.01443083663578714303722357226, 2.87277240523447263591486231079, 3.55092888160363787798225954509, 4.91600442197387406585931308504, 5.48456408092783435616763838861, 6.01636785271938578895132479284, 7.28018757441796852813299079351, 7.62588663573730151592930986823, 8.596625433208947405879267301543
