| L(s) = 1 | + (−0.123 − 1.40i)2-s + (−1.11 − 1.32i)3-s + (−1.96 + 0.347i)4-s + (1.21 − 0.564i)5-s + (−1.73 + 1.73i)6-s + (−1.35 − 0.492i)7-s + (0.732 + 2.73i)8-s + (−0.520 + 2.95i)9-s + (−0.945 − 1.63i)10-s + (−16.7 − 9.66i)11-s + (2.65 + 2.22i)12-s + (−11.4 − 7.99i)13-s + (−0.527 + 1.96i)14-s + (−2.09 − 0.978i)15-s + (3.75 − 1.36i)16-s + (4.56 + 6.52i)17-s + ⋯ |
| L(s) = 1 | + (−0.0616 − 0.704i)2-s + (−0.371 − 0.442i)3-s + (−0.492 + 0.0868i)4-s + (0.242 − 0.112i)5-s + (−0.288 + 0.288i)6-s + (−0.193 − 0.0703i)7-s + (0.0915 + 0.341i)8-s + (−0.0578 + 0.328i)9-s + (−0.0945 − 0.163i)10-s + (−1.52 − 0.878i)11-s + (0.221 + 0.185i)12-s + (−0.878 − 0.614i)13-s + (−0.0376 + 0.140i)14-s + (−0.139 − 0.0652i)15-s + (0.234 − 0.0855i)16-s + (0.268 + 0.383i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 222 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 222 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.116189 + 0.291855i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.116189 + 0.291855i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.123 + 1.40i)T \) |
| 3 | \( 1 + (1.11 + 1.32i)T \) |
| 37 | \( 1 + (3.94 + 36.7i)T \) |
| good | 5 | \( 1 + (-1.21 + 0.564i)T + (16.0 - 19.1i)T^{2} \) |
| 7 | \( 1 + (1.35 + 0.492i)T + (37.5 + 31.4i)T^{2} \) |
| 11 | \( 1 + (16.7 + 9.66i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (11.4 + 7.99i)T + (57.8 + 158. i)T^{2} \) |
| 17 | \( 1 + (-4.56 - 6.52i)T + (-98.8 + 271. i)T^{2} \) |
| 19 | \( 1 + (3.10 - 35.4i)T + (-355. - 62.6i)T^{2} \) |
| 23 | \( 1 + (13.6 + 3.66i)T + (458. + 264.5i)T^{2} \) |
| 29 | \( 1 + (-36.0 + 9.64i)T + (728. - 420.5i)T^{2} \) |
| 31 | \( 1 + (26.7 + 26.7i)T + 961iT^{2} \) |
| 41 | \( 1 + (13.6 - 2.40i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (32.1 - 32.1i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (21.5 + 37.3i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-69.2 + 25.2i)T + (2.15e3 - 1.80e3i)T^{2} \) |
| 59 | \( 1 + (-30.9 + 66.3i)T + (-2.23e3 - 2.66e3i)T^{2} \) |
| 61 | \( 1 + (43.3 - 61.9i)T + (-1.27e3 - 3.49e3i)T^{2} \) |
| 67 | \( 1 + (-40.7 + 112. i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-9.05 + 7.59i)T + (875. - 4.96e3i)T^{2} \) |
| 73 | \( 1 + 37.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (48.2 - 22.5i)T + (4.01e3 - 4.78e3i)T^{2} \) |
| 83 | \( 1 + (3.44 - 19.5i)T + (-6.47e3 - 2.35e3i)T^{2} \) |
| 89 | \( 1 + (65.2 + 30.4i)T + (5.09e3 + 6.06e3i)T^{2} \) |
| 97 | \( 1 + (-161. - 43.1i)T + (8.14e3 + 4.70e3i)T^{2} \) |
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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
−11.49119604663327761017627166551, −10.36372631892519581048346936485, −9.953138253370140111579502174960, −8.295637197200951004686606768370, −7.70834882262354445166704567358, −5.97911761103335558088351765192, −5.21727692432183898012359948481, −3.48685805935875962780368734788, −2.07632322213144784660169656640, −0.17166532507812453412717598848,
2.63553102544738937054011443531, 4.61510922078739243166328521928, 5.23121821612182385855981226768, 6.63924138050743613488203138723, 7.44237973506936764851361321001, 8.732849277738913211985206918997, 9.834323525142658291169658399608, 10.36930830019138981431444060779, 11.71923081935543254773406759758, 12.70369155322407409016307685265
