| L(s) = 1 | − 5.19·7-s + 25.9i·13-s − 11i·19-s − 25·25-s + 41.5·31-s − 57.1i·37-s − 22i·43-s − 22·49-s + 15.5i·61-s − 109i·67-s − 97·73-s − 88.3·79-s − 135i·91-s − 167·97-s − 202.·103-s + ⋯ |
| L(s) = 1 | − 0.742·7-s + 1.99i·13-s − 0.578i·19-s − 25-s + 1.34·31-s − 1.54i·37-s − 0.511i·43-s − 0.448·49-s + 0.255i·61-s − 1.62i·67-s − 1.32·73-s − 1.11·79-s − 1.48i·91-s − 1.72·97-s − 1.96·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3732211133\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3732211133\) |
| \(L(2)\) |
|
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 | \( 1 + 25T^{2} \) |
| 7 | \( 1 + 5.19T + 49T^{2} \) |
| 11 | \( 1 + 121T^{2} \) |
| 13 | \( 1 - 25.9iT - 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 + 11iT - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 + 841T^{2} \) |
| 31 | \( 1 - 41.5T + 961T^{2} \) |
| 37 | \( 1 + 57.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 + 22iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 + 3.48e3T^{2} \) |
| 61 | \( 1 - 15.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 109iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 97T + 5.32e3T^{2} \) |
| 79 | \( 1 + 88.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 + 167T + 9.40e3T^{2} \) |
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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
−9.091974792854938213762606460431, −8.071227886419792927410877309836, −7.05718270965819789724448548796, −6.55944772872370151915746165443, −5.70577537305413716012082877739, −4.52717429702068371643725573501, −3.89220453466778426556960356743, −2.70962216530542058294491793837, −1.69078622287830428976148504395, −0.10374960372011467261098378236,
1.15977788631638934012281273204, 2.72829059021338353600116407169, 3.34481484479885725709132914028, 4.45763092661184729573160644324, 5.55101890335721145547067257816, 6.10070939398177642624342968401, 7.04760398391604279700625649585, 8.045046784222922393716493676610, 8.390136592826943897420887786259, 9.766301496085489235299881725955
