| L(s) = 1 | + 3.30·3-s + 5-s − 0.302·7-s + 7.90·9-s − 3.30·11-s + 2.30·13-s + 3.30·15-s + 7.30·17-s + 2.90·19-s − 1.00·21-s − 23-s + 25-s + 16.2·27-s − 8.60·29-s + 4.30·31-s − 10.9·33-s − 0.302·35-s − 5.21·37-s + 7.60·39-s − 4.69·41-s + 4·43-s + 7.90·45-s + 3.39·47-s − 6.90·49-s + 24.1·51-s + 7.21·53-s − 3.30·55-s + ⋯ |
| L(s) = 1 | + 1.90·3-s + 0.447·5-s − 0.114·7-s + 2.63·9-s − 0.995·11-s + 0.638·13-s + 0.852·15-s + 1.77·17-s + 0.667·19-s − 0.218·21-s − 0.208·23-s + 0.200·25-s + 3.11·27-s − 1.59·29-s + 0.772·31-s − 1.89·33-s − 0.0511·35-s − 0.856·37-s + 1.21·39-s − 0.733·41-s + 0.609·43-s + 1.17·45-s + 0.495·47-s − 0.986·49-s + 3.37·51-s + 0.990·53-s − 0.445·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.185204103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.185204103\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 3.30T + 3T^{2} \) |
| 7 | \( 1 + 0.302T + 7T^{2} \) |
| 11 | \( 1 + 3.30T + 11T^{2} \) |
| 13 | \( 1 - 2.30T + 13T^{2} \) |
| 17 | \( 1 - 7.30T + 17T^{2} \) |
| 19 | \( 1 - 2.90T + 19T^{2} \) |
| 29 | \( 1 + 8.60T + 29T^{2} \) |
| 31 | \( 1 - 4.30T + 31T^{2} \) |
| 37 | \( 1 + 5.21T + 37T^{2} \) |
| 41 | \( 1 + 4.69T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 3.39T + 47T^{2} \) |
| 53 | \( 1 - 7.21T + 53T^{2} \) |
| 59 | \( 1 - 1.39T + 59T^{2} \) |
| 61 | \( 1 + 1.30T + 61T^{2} \) |
| 67 | \( 1 - 9.21T + 67T^{2} \) |
| 71 | \( 1 - 5.30T + 71T^{2} \) |
| 73 | \( 1 - 5.39T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 2.78T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−8.112662492387008552132388305765, −7.43370896929000770674778093950, −6.80900060976374921837268552928, −5.67421924558731445474856511183, −5.12437900245298959419213104552, −3.94747818359360939156910204163, −3.41695131955969658481557877061, −2.78932159777582304299458928396, −1.97847992146891655027362243393, −1.13346635341198203923847893046,
1.13346635341198203923847893046, 1.97847992146891655027362243393, 2.78932159777582304299458928396, 3.41695131955969658481557877061, 3.94747818359360939156910204163, 5.12437900245298959419213104552, 5.67421924558731445474856511183, 6.80900060976374921837268552928, 7.43370896929000770674778093950, 8.112662492387008552132388305765
