| L(s) = 1 | − 0.533·2-s + 3.38·3-s − 1.71·4-s − 0.716·5-s − 1.80·6-s − 4.66·7-s + 1.98·8-s + 8.42·9-s + 0.382·10-s + 3.05·11-s − 5.79·12-s + 4.41·13-s + 2.49·14-s − 2.42·15-s + 2.37·16-s − 17-s − 4.49·18-s + 5.07·19-s + 1.22·20-s − 15.7·21-s − 1.63·22-s + 3.55·23-s + 6.70·24-s − 4.48·25-s − 2.35·26-s + 18.3·27-s + 8.00·28-s + ⋯ |
| L(s) = 1 | − 0.377·2-s + 1.95·3-s − 0.857·4-s − 0.320·5-s − 0.736·6-s − 1.76·7-s + 0.701·8-s + 2.80·9-s + 0.121·10-s + 0.921·11-s − 1.67·12-s + 1.22·13-s + 0.666·14-s − 0.625·15-s + 0.592·16-s − 0.242·17-s − 1.06·18-s + 1.16·19-s + 0.274·20-s − 3.44·21-s − 0.347·22-s + 0.740·23-s + 1.36·24-s − 0.897·25-s − 0.462·26-s + 3.52·27-s + 1.51·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.895433161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.895433161\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 0.533T + 2T^{2} \) |
| 3 | \( 1 - 3.38T + 3T^{2} \) |
| 5 | \( 1 + 0.716T + 5T^{2} \) |
| 7 | \( 1 + 4.66T + 7T^{2} \) |
| 11 | \( 1 - 3.05T + 11T^{2} \) |
| 13 | \( 1 - 4.41T + 13T^{2} \) |
| 19 | \( 1 - 5.07T + 19T^{2} \) |
| 23 | \( 1 - 3.55T + 23T^{2} \) |
| 29 | \( 1 + 6.09T + 29T^{2} \) |
| 31 | \( 1 - 7.42T + 31T^{2} \) |
| 37 | \( 1 + 1.08T + 37T^{2} \) |
| 41 | \( 1 - 8.24T + 41T^{2} \) |
| 43 | \( 1 - 4.36T + 43T^{2} \) |
| 53 | \( 1 + 2.88T + 53T^{2} \) |
| 59 | \( 1 + 6.93T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 - 0.474T + 67T^{2} \) |
| 71 | \( 1 - 0.978T + 71T^{2} \) |
| 73 | \( 1 + 8.55T + 73T^{2} \) |
| 79 | \( 1 - 5.39T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 + 4.47T + 97T^{2} \) |
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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
−9.627964759507135684863391930663, −9.366126547323631130660236340021, −8.848714385374680830883947956741, −7.901645363064046683894321020833, −7.17938197783799039762769192424, −6.11271586097084947347351114907, −4.26188279863991578895761609809, −3.65465982831335282760539525518, −2.98357853619639631502033569053, −1.22670160348972961889014829040,
1.22670160348972961889014829040, 2.98357853619639631502033569053, 3.65465982831335282760539525518, 4.26188279863991578895761609809, 6.11271586097084947347351114907, 7.17938197783799039762769192424, 7.901645363064046683894321020833, 8.848714385374680830883947956741, 9.366126547323631130660236340021, 9.627964759507135684863391930663
