| L(s) = 1 | + 5-s + 4.35i·7-s − 3·9-s − 4.35i·11-s + 7·17-s + 4.35i·19-s + 8.71i·23-s − 4·25-s + 4.35i·35-s + 13.0i·43-s − 3·45-s + 4.35i·47-s − 12.0·49-s − 4.35i·55-s − 15·61-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.64i·7-s − 9-s − 1.31i·11-s + 1.69·17-s + 0.999i·19-s + 1.81i·23-s − 0.800·25-s + 0.736i·35-s + 1.99i·43-s − 0.447·45-s + 0.635i·47-s − 1.71·49-s − 0.587i·55-s − 1.92·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.419278522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.419278522\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 - 4.35iT - 7T^{2} \) |
| 11 | \( 1 + 4.35iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 7T + 17T^{2} \) |
| 23 | \( 1 - 8.71iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 13.0iT - 43T^{2} \) |
| 47 | \( 1 - 4.35iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 15T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 8.71iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.598640470651873021927810079484, −9.289689289756409509468176085632, −8.182529881711780159313974762446, −7.892461696995965956192970964650, −6.05933519784711203569568899208, −5.84570053584007553397729896467, −5.27621839607647452944859918194, −3.44894845950579846685311982810, −2.88082250231297329570457088281, −1.54550165955808104194140134970,
0.61090724662325347051808232074, 2.10492350651637610444647579425, 3.33993944117661109114552992065, 4.36506464712612395828591044825, 5.19137696551245104377824290314, 6.27600795084745381564055865545, 7.14724539298870820808912222030, 7.72824453972356422499019613261, 8.730652112905267376001911698192, 9.733159759828253484517382507585
