| L(s) = 1 | − 3-s − 4i·5-s − 2i·7-s + 9-s − 2i·11-s + (−3 − 2i)13-s + 4i·15-s − 6·17-s − 2i·19-s + 2i·21-s + 8·23-s − 11·25-s − 27-s − 6·29-s − 10i·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78i·5-s − 0.755i·7-s + 0.333·9-s − 0.603i·11-s + (−0.832 − 0.554i)13-s + 1.03i·15-s − 1.45·17-s − 0.458i·19-s + 0.436i·21-s + 1.66·23-s − 2.20·25-s − 0.192·27-s − 1.11·29-s − 1.79i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7797835954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7797835954\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + (3 + 2i)T \) |
| good | 5 | \( 1 + 4iT - 5T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 10iT - 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 - 14iT - 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 6iT - 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 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
−8.649102594500515562342702447315, −7.65050476722099170218013605452, −7.04766700526527336053534336504, −5.91509934996935650689321755728, −5.24115070357152327931453580061, −4.56581713823164164053018756248, −3.95039842533046480741083629369, −2.41499891062322331168573727567, −1.03603562996666418277480088409, −0.31966947271125113613729335349,
1.98147274487239848904423483903, 2.62640295538608000878032812571, 3.65988317916029700707079890199, 4.72524061982734275790183561088, 5.55676552435834742565901160497, 6.50113669118761307461248035513, 7.00768809896649611309691116990, 7.42529433920466708307214577013, 8.766921861596550770230391365524, 9.430186903857130543204243077918
