| L(s) = 1 | − 19.7i·5-s + 34·7-s − 5.65i·11-s − 267·25-s − 223. i·29-s + 70·31-s − 673. i·35-s + 813·49-s − 579. i·53-s − 112.·55-s + 554. i·59-s − 322·73-s − 192. i·77-s − 1.37e3·79-s − 1.22e3i·83-s + ⋯ |
| L(s) = 1 | − 1.77i·5-s + 1.83·7-s − 0.155i·11-s − 2.13·25-s − 1.43i·29-s + 0.405·31-s − 3.25i·35-s + 2.37·49-s − 1.50i·53-s − 0.274·55-s + 1.22i·59-s − 0.516·73-s − 0.284i·77-s − 1.95·79-s − 1.62i·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.108919687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.108919687\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 19.7iT - 125T^{2} \) |
| 7 | \( 1 - 34T + 343T^{2} \) |
| 11 | \( 1 + 5.65iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 2.19e3T^{2} \) |
| 17 | \( 1 + 4.91e3T^{2} \) |
| 19 | \( 1 - 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 + 223. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 70T + 2.97e4T^{2} \) |
| 37 | \( 1 - 5.06e4T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 - 7.95e4T^{2} \) |
| 47 | \( 1 + 1.03e5T^{2} \) |
| 53 | \( 1 + 579. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 554. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 2.26e5T^{2} \) |
| 67 | \( 1 - 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 + 322T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.37e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.22e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 + 574T + 9.12e5T^{2} \) |
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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
−11.48214944191593294086476533915, −10.15296807908926188882202869298, −8.955815583917076791416776789525, −8.334164412457794528064257586476, −7.63005488804058618439352639941, −5.79052724918491123696966831234, −4.89951563441063662155692358808, −4.23736571867318702157920557289, −1.93407725571111384367408329568, −0.854692854043478836273477347845,
1.73220427893793231135780452322, 2.97176518867559875544422242109, 4.38150302547969145180981064832, 5.62391519078560818375936434318, 6.89675995251773808252489858093, 7.60638010169990584021963497024, 8.573577037459673877182630969198, 10.00044265947340884484848528676, 10.93236559115647659812227445442, 11.24228682408501507032598225570
