| L(s) = 1 | + 3.27·5-s − 0.418i·7-s − 3·9-s − 6.50i·11-s − 7.27·17-s − 4.35i·19-s − 8.71i·23-s + 5.72·25-s − 1.37i·35-s + 5.67i·43-s − 9.82·45-s + 13.4i·47-s + 6.82·49-s − 21.3i·55-s + 11.2·61-s + ⋯ |
| L(s) = 1 | + 1.46·5-s − 0.158i·7-s − 9-s − 1.96i·11-s − 1.76·17-s − 0.999i·19-s − 1.81i·23-s + 1.14·25-s − 0.231i·35-s + 0.865i·43-s − 1.46·45-s + 1.96i·47-s + 0.974·49-s − 2.87i·55-s + 1.44·61-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
| \(L(1)\) |
\(\approx\) |
\(1.597704003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.597704003\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.27T + 5T^{2} \) |
| 7 | \( 1 + 0.418iT - 7T^{2} \) |
| 11 | \( 1 + 6.50iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 7.27T + 17T^{2} \) |
| 23 | \( 1 + 8.71iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 5.67iT - 43T^{2} \) |
| 47 | \( 1 - 13.4iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 5.82T + 73T^{2} \) |
| 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
−9.259568933484379987769908533645, −8.862546313361016445414075410385, −8.207140556315168902042586008085, −6.62695190903962523614635600408, −6.23638268251933107140580362478, −5.47996785995750485803002956296, −4.46217021859284210959720279501, −2.95131376303233461609682051183, −2.35283976722074823095843499678, −0.63074735473076003572080596253,
1.88869366439457487855130310765, 2.29995058278422064440349005770, 3.84917689826698829926305016667, 5.07908684880304459973525029547, 5.64201792171364101788997702964, 6.59112609498170056735028018995, 7.30993388739700851338498588213, 8.512145742406747764837561753160, 9.275581923933141761895357381464, 9.831980300858860564174184011178
