| L(s) = 1 | + (−3.25 − 2.36i)2-s + (0.539 + 5.13i)3-s + (2.53 + 7.81i)4-s + (4.24 + 7.34i)5-s + (10.3 − 18.0i)6-s + (22.3 + 4.75i)7-s + (0.271 − 0.834i)8-s + (0.327 − 0.0696i)9-s + (3.57 − 33.9i)10-s + (−35.7 + 39.7i)11-s + (−38.7 + 17.2i)12-s + (−18.5 − 8.25i)13-s + (−61.5 − 68.3i)14-s + (−35.4 + 25.7i)15-s + (50.3 − 36.5i)16-s + (9.22 + 10.2i)17-s + ⋯ |
| L(s) = 1 | + (−1.15 − 0.836i)2-s + (0.103 + 0.988i)3-s + (0.317 + 0.976i)4-s + (0.379 + 0.657i)5-s + (0.707 − 1.22i)6-s + (1.20 + 0.256i)7-s + (0.0119 − 0.0368i)8-s + (0.0121 − 0.00257i)9-s + (0.112 − 1.07i)10-s + (−0.980 + 1.08i)11-s + (−0.932 + 0.415i)12-s + (−0.395 − 0.176i)13-s + (−1.17 − 1.30i)14-s + (−0.609 + 0.443i)15-s + (0.786 − 0.571i)16-s + (0.131 + 0.146i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.843 - 0.536i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.843 - 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.761172 + 0.221567i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.761172 + 0.221567i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + (-7.52 - 172. i)T \) |
| good | 2 | \( 1 + (3.25 + 2.36i)T + (2.47 + 7.60i)T^{2} \) |
| 3 | \( 1 + (-0.539 - 5.13i)T + (-26.4 + 5.61i)T^{2} \) |
| 5 | \( 1 + (-4.24 - 7.34i)T + (-62.5 + 108. i)T^{2} \) |
| 7 | \( 1 + (-22.3 - 4.75i)T + (313. + 139. i)T^{2} \) |
| 11 | \( 1 + (35.7 - 39.7i)T + (-139. - 1.32e3i)T^{2} \) |
| 13 | \( 1 + (18.5 + 8.25i)T + (1.47e3 + 1.63e3i)T^{2} \) |
| 17 | \( 1 + (-9.22 - 10.2i)T + (-513. + 4.88e3i)T^{2} \) |
| 19 | \( 1 + (-22.1 + 9.85i)T + (4.58e3 - 5.09e3i)T^{2} \) |
| 23 | \( 1 + (-57.4 + 176. i)T + (-9.84e3 - 7.15e3i)T^{2} \) |
| 29 | \( 1 + (172. + 125. i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 37 | \( 1 + (-101. + 175. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (16.6 - 158. i)T + (-6.74e4 - 1.43e4i)T^{2} \) |
| 43 | \( 1 + (-172. + 77.0i)T + (5.32e4 - 5.90e4i)T^{2} \) |
| 47 | \( 1 + (-425. + 309. i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (153. - 32.7i)T + (1.36e5 - 6.05e4i)T^{2} \) |
| 59 | \( 1 + (-46.2 - 440. i)T + (-2.00e5 + 4.27e4i)T^{2} \) |
| 61 | \( 1 + 770.T + 2.26e5T^{2} \) |
| 67 | \( 1 + (375. + 651. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-1.11e3 + 236. i)T + (3.26e5 - 1.45e5i)T^{2} \) |
| 73 | \( 1 + (-323. + 359. i)T + (-4.06e4 - 3.86e5i)T^{2} \) |
| 79 | \( 1 + (-151. - 167. i)T + (-5.15e4 + 4.90e5i)T^{2} \) |
| 83 | \( 1 + (-67.3 + 640. i)T + (-5.59e5 - 1.18e5i)T^{2} \) |
| 89 | \( 1 + (159. + 490. i)T + (-5.70e5 + 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-155. - 478. i)T + (-7.38e5 + 5.36e5i)T^{2} \) |
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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
−16.89487930137963271010299722087, −15.25397574417610777134967424669, −14.51069671920644978589944189406, −12.35575827254976863253949988931, −10.79073596712953526293575240427, −10.33299874803565070015334129311, −9.142654560935373816277084609954, −7.67632150157530521992062323072, −4.90794171965234477680718969654, −2.36107263406277292360436353512,
1.23537171081669143274545178777, 5.52378442010458085156278877261, 7.40280948673912634354127590881, 8.055747781746135608372926644165, 9.372186723840129488111387989768, 11.08657926424452472862073139662, 12.82979014057840442873730893007, 13.86548345843798514637222242419, 15.43892145998581748680609114953, 16.71107991721442957605467363732
