| L(s) = 1 | + 1.44·5-s + 7-s + 2·11-s + 4.89·13-s − 2·17-s − 2.55·19-s − 23-s − 2.89·25-s + 6.89·29-s − 6·31-s + 1.44·35-s + 11.7·37-s − 9.79·41-s − 6.89·43-s + 9.79·47-s + 49-s + 10.8·53-s + 2.89·55-s − 2·59-s + 6.55·61-s + 7.10·65-s + 12.8·67-s + 0.101·71-s − 6.89·73-s + 2·77-s + 1.89·79-s + 2·83-s + ⋯ |
| L(s) = 1 | + 0.648·5-s + 0.377·7-s + 0.603·11-s + 1.35·13-s − 0.485·17-s − 0.585·19-s − 0.208·23-s − 0.579·25-s + 1.28·29-s − 1.07·31-s + 0.245·35-s + 1.93·37-s − 1.53·41-s − 1.05·43-s + 1.42·47-s + 0.142·49-s + 1.49·53-s + 0.390·55-s − 0.260·59-s + 0.838·61-s + 0.880·65-s + 1.57·67-s + 0.0119·71-s − 0.807·73-s + 0.227·77-s + 0.213·79-s + 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.827274040\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.827274040\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 1.44T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 4.89T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 2.55T + 19T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 - 6.89T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 + 9.79T + 41T^{2} \) |
| 43 | \( 1 + 6.89T + 43T^{2} \) |
| 47 | \( 1 - 9.79T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + 2T + 59T^{2} \) |
| 61 | \( 1 - 6.55T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 - 0.101T + 71T^{2} \) |
| 73 | \( 1 + 6.89T + 73T^{2} \) |
| 79 | \( 1 - 1.89T + 79T^{2} \) |
| 83 | \( 1 - 2T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 - 2.89T + 97T^{2} \) |
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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
−7.82038464544903319263206966495, −6.87081939073847648021393907801, −6.33257667762718718182413818760, −5.80495920410502349463955589595, −5.00412268718490120571879899049, −4.12736257027854662369664200828, −3.61614667015159819703147009218, −2.45550484320101668634316809966, −1.75874268382395736534777293173, −0.847117957674979075798855078951,
0.847117957674979075798855078951, 1.75874268382395736534777293173, 2.45550484320101668634316809966, 3.61614667015159819703147009218, 4.12736257027854662369664200828, 5.00412268718490120571879899049, 5.80495920410502349463955589595, 6.33257667762718718182413818760, 6.87081939073847648021393907801, 7.82038464544903319263206966495
