| L(s) = 1 | + 2.58·2-s + 0.884·3-s + 4.68·4-s − 5-s + 2.28·6-s − 0.858·7-s + 6.95·8-s − 2.21·9-s − 2.58·10-s + 6.21·11-s + 4.14·12-s − 2.22·14-s − 0.884·15-s + 8.61·16-s + 3.95·17-s − 5.73·18-s − 6.34·19-s − 4.68·20-s − 0.759·21-s + 16.0·22-s + 3.80·23-s + 6.15·24-s + 25-s − 4.61·27-s − 4.02·28-s + 0.142·29-s − 2.28·30-s + ⋯ |
| L(s) = 1 | + 1.82·2-s + 0.510·3-s + 2.34·4-s − 0.447·5-s + 0.934·6-s − 0.324·7-s + 2.45·8-s − 0.739·9-s − 0.817·10-s + 1.87·11-s + 1.19·12-s − 0.593·14-s − 0.228·15-s + 2.15·16-s + 0.959·17-s − 1.35·18-s − 1.45·19-s − 1.04·20-s − 0.165·21-s + 3.42·22-s + 0.792·23-s + 1.25·24-s + 0.200·25-s − 0.888·27-s − 0.760·28-s + 0.0265·29-s − 0.417·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.951157182\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.951157182\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.58T + 2T^{2} \) |
| 3 | \( 1 - 0.884T + 3T^{2} \) |
| 7 | \( 1 + 0.858T + 7T^{2} \) |
| 11 | \( 1 - 6.21T + 11T^{2} \) |
| 17 | \( 1 - 3.95T + 17T^{2} \) |
| 19 | \( 1 + 6.34T + 19T^{2} \) |
| 23 | \( 1 - 3.80T + 23T^{2} \) |
| 29 | \( 1 - 0.142T + 29T^{2} \) |
| 31 | \( 1 + 5.21T + 31T^{2} \) |
| 37 | \( 1 + 9.04T + 37T^{2} \) |
| 41 | \( 1 + 2.38T + 41T^{2} \) |
| 43 | \( 1 - 4.48T + 43T^{2} \) |
| 47 | \( 1 + 9.92T + 47T^{2} \) |
| 53 | \( 1 - 5.45T + 53T^{2} \) |
| 59 | \( 1 - 3.55T + 59T^{2} \) |
| 61 | \( 1 + 2.92T + 61T^{2} \) |
| 67 | \( 1 + 3.98T + 67T^{2} \) |
| 71 | \( 1 + 1.45T + 71T^{2} \) |
| 73 | \( 1 + 5.61T + 73T^{2} \) |
| 79 | \( 1 - 0.550T + 79T^{2} \) |
| 83 | \( 1 + 4.27T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 - 4.50T + 97T^{2} \) |
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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
−10.56554983802791780590250959413, −9.242627935418672751823187499767, −8.443257471707140237658523357834, −7.22233624914589921185532864076, −6.52011202563548096945372758929, −5.74794853936200539130317313045, −4.63976449273553048065144455653, −3.66911889654197713400473279392, −3.23740158686382494210857413071, −1.84748794880216744432625900030,
1.84748794880216744432625900030, 3.23740158686382494210857413071, 3.66911889654197713400473279392, 4.63976449273553048065144455653, 5.74794853936200539130317313045, 6.52011202563548096945372758929, 7.22233624914589921185532864076, 8.443257471707140237658523357834, 9.242627935418672751823187499767, 10.56554983802791780590250959413
