| L(s) = 1 | + 3-s − 4.68i·7-s + 9-s − 2.29i·11-s − 4.97·13-s − 2.97i·17-s + 2.68i·19-s − 4.68i·21-s + 2.68i·23-s + 27-s − 2i·29-s + 6.97·31-s − 2.29i·33-s − 4.39·37-s − 4.97·39-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.77i·7-s + 0.333·9-s − 0.691i·11-s − 1.38·13-s − 0.722i·17-s + 0.616i·19-s − 1.02i·21-s + 0.560i·23-s + 0.192·27-s − 0.371i·29-s + 1.25·31-s − 0.399i·33-s − 0.722·37-s − 0.797·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.818 + 0.574i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.818 + 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.299267800\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.299267800\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4.68iT - 7T^{2} \) |
| 11 | \( 1 + 2.29iT - 11T^{2} \) |
| 13 | \( 1 + 4.97T + 13T^{2} \) |
| 17 | \( 1 + 2.97iT - 17T^{2} \) |
| 19 | \( 1 - 2.68iT - 19T^{2} \) |
| 23 | \( 1 - 2.68iT - 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 - 6.97T + 31T^{2} \) |
| 37 | \( 1 + 4.39T + 37T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 + 9.37T + 43T^{2} \) |
| 47 | \( 1 + 7.27iT - 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 1.70iT - 59T^{2} \) |
| 61 | \( 1 - 4.58iT - 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 0.585T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 1.02T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 + 3.37T + 89T^{2} \) |
| 97 | \( 1 + 3.95iT - 97T^{2} \) |
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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
−8.456451703298556878211931864397, −7.911490432516477481752008925124, −7.09851664667425053076451978662, −6.70908162748921289147044402792, −5.32441083380088521377280602965, −4.56414167715133082915611853668, −3.68607614862327999040566169900, −2.96919658027631263968802931052, −1.64652685770277551560491194768, −0.37674848081668442111708560378,
1.86004755616638252934571693435, 2.49583725700872458240328398898, 3.32996823022288471911372854979, 4.78261303103634913302310500543, 5.07403073849512085859444342013, 6.29228541305181856156608583893, 6.90669624517654936236125367643, 7.973916564848626908365903408775, 8.499524715562551454341702302506, 9.239743367561222786025352361069
