| L(s) = 1 | + 1.80·3-s + 3.12·5-s + 4.44·7-s + 0.246·9-s + 4.76·11-s + 1.14·13-s + 5.62·15-s − 4.01·17-s + 4.12·19-s + 8.01·21-s − 6.38·23-s + 4.74·25-s − 4.96·27-s + 3.55·29-s + 0.958·31-s + 8.58·33-s + 13.8·35-s − 8.42·37-s + 2.05·39-s − 6.44·41-s + 0.770·45-s + 8.47·47-s + 12.7·49-s − 7.22·51-s + 7.42·53-s + 14.8·55-s + 7.43·57-s + ⋯ |
| L(s) = 1 | + 1.04·3-s + 1.39·5-s + 1.68·7-s + 0.0823·9-s + 1.43·11-s + 0.316·13-s + 1.45·15-s − 0.972·17-s + 0.946·19-s + 1.74·21-s − 1.33·23-s + 0.948·25-s − 0.954·27-s + 0.659·29-s + 0.172·31-s + 1.49·33-s + 2.34·35-s − 1.38·37-s + 0.329·39-s − 1.00·41-s + 0.114·45-s + 1.23·47-s + 1.82·49-s − 1.01·51-s + 1.02·53-s + 2.00·55-s + 0.985·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7396 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.372966707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.372966707\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 \) |
| good | 3 | \( 1 - 1.80T + 3T^{2} \) |
| 5 | \( 1 - 3.12T + 5T^{2} \) |
| 7 | \( 1 - 4.44T + 7T^{2} \) |
| 11 | \( 1 - 4.76T + 11T^{2} \) |
| 13 | \( 1 - 1.14T + 13T^{2} \) |
| 17 | \( 1 + 4.01T + 17T^{2} \) |
| 19 | \( 1 - 4.12T + 19T^{2} \) |
| 23 | \( 1 + 6.38T + 23T^{2} \) |
| 29 | \( 1 - 3.55T + 29T^{2} \) |
| 31 | \( 1 - 0.958T + 31T^{2} \) |
| 37 | \( 1 + 8.42T + 37T^{2} \) |
| 41 | \( 1 + 6.44T + 41T^{2} \) |
| 47 | \( 1 - 8.47T + 47T^{2} \) |
| 53 | \( 1 - 7.42T + 53T^{2} \) |
| 59 | \( 1 + 3.92T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 + 1.39T + 71T^{2} \) |
| 73 | \( 1 - 3.94T + 73T^{2} \) |
| 79 | \( 1 + 8.30T + 79T^{2} \) |
| 83 | \( 1 - 1.84T + 83T^{2} \) |
| 89 | \( 1 - 6.84T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−8.086026961991568814809928232400, −7.29737892450072390452092314416, −6.48850684153582219162755627610, −5.79552464156106758754715367637, −5.10125632334767641620313184440, −4.27330956889273893618821260772, −3.54055422087001098536094228098, −2.41479088970236538930685091976, −1.86096405738752370231649046208, −1.28701781711546068677404590687,
1.28701781711546068677404590687, 1.86096405738752370231649046208, 2.41479088970236538930685091976, 3.54055422087001098536094228098, 4.27330956889273893618821260772, 5.10125632334767641620313184440, 5.79552464156106758754715367637, 6.48850684153582219162755627610, 7.29737892450072390452092314416, 8.086026961991568814809928232400
