| L(s) = 1 | + 1.99·2-s + 1.96·4-s + 3.87·5-s + 4.52·7-s − 0.0695·8-s + 7.70·10-s + 2.70·11-s − 4.10·13-s + 9.01·14-s − 4.06·16-s − 0.582·17-s + 7.60·20-s + 5.39·22-s − 1.41·23-s + 9.97·25-s − 8.18·26-s + 8.89·28-s + 5.61·29-s + 0.639·31-s − 7.96·32-s − 1.15·34-s + 17.5·35-s − 6.21·37-s − 0.269·40-s + 1.77·41-s + 4.91·43-s + 5.32·44-s + ⋯ |
| L(s) = 1 | + 1.40·2-s + 0.982·4-s + 1.73·5-s + 1.71·7-s − 0.0245·8-s + 2.43·10-s + 0.816·11-s − 1.13·13-s + 2.40·14-s − 1.01·16-s − 0.141·17-s + 1.70·20-s + 1.15·22-s − 0.294·23-s + 1.99·25-s − 1.60·26-s + 1.68·28-s + 1.04·29-s + 0.114·31-s − 1.40·32-s − 0.198·34-s + 2.96·35-s − 1.02·37-s − 0.0425·40-s + 0.276·41-s + 0.748·43-s + 0.802·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.372269516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.372269516\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 1.99T + 2T^{2} \) |
| 5 | \( 1 - 3.87T + 5T^{2} \) |
| 7 | \( 1 - 4.52T + 7T^{2} \) |
| 11 | \( 1 - 2.70T + 11T^{2} \) |
| 13 | \( 1 + 4.10T + 13T^{2} \) |
| 17 | \( 1 + 0.582T + 17T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 - 5.61T + 29T^{2} \) |
| 31 | \( 1 - 0.639T + 31T^{2} \) |
| 37 | \( 1 + 6.21T + 37T^{2} \) |
| 41 | \( 1 - 1.77T + 41T^{2} \) |
| 43 | \( 1 - 4.91T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 8.50T + 53T^{2} \) |
| 59 | \( 1 - 2.46T + 59T^{2} \) |
| 61 | \( 1 + 7.82T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 - 3.72T + 71T^{2} \) |
| 73 | \( 1 + 3.23T + 73T^{2} \) |
| 79 | \( 1 + 3.96T + 79T^{2} \) |
| 83 | \( 1 + 2.13T + 83T^{2} \) |
| 89 | \( 1 - 17.1T + 89T^{2} \) |
| 97 | \( 1 - 9.58T + 97T^{2} \) |
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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
−8.775570420854638724029192822559, −7.76192727945854397829023114851, −6.77373571059650581875174224690, −6.17461322818634772032895300226, −5.38691441938439936202251937422, −4.85939937097298042608952761908, −4.36180533963334714999408128066, −3.01367215890316774864273319148, −2.14311460649025784357257139121, −1.50234153519075289354104983459,
1.50234153519075289354104983459, 2.14311460649025784357257139121, 3.01367215890316774864273319148, 4.36180533963334714999408128066, 4.85939937097298042608952761908, 5.38691441938439936202251937422, 6.17461322818634772032895300226, 6.77373571059650581875174224690, 7.76192727945854397829023114851, 8.775570420854638724029192822559
