| L(s) = 1 | + (−0.199 − 0.209i)2-s + (−2.27 − 0.852i)3-s + (0.0859 − 1.91i)4-s + (−0.567 − 0.411i)5-s + (0.275 + 0.645i)6-s + (0.218 − 1.61i)7-s + (−0.852 + 0.745i)8-s + (2.17 + 1.90i)9-s + (0.0272 + 0.200i)10-s + (1.78 − 1.06i)11-s + (−1.82 + 4.27i)12-s + (3.86 + 2.30i)13-s + (−0.380 + 0.276i)14-s + (0.937 + 1.41i)15-s + (−3.49 − 0.314i)16-s + (−1.45 + 4.48i)17-s + ⋯ |
| L(s) = 1 | + (−0.141 − 0.147i)2-s + (−1.31 − 0.492i)3-s + (0.0429 − 0.957i)4-s + (−0.253 − 0.184i)5-s + (0.112 + 0.263i)6-s + (0.0824 − 0.609i)7-s + (−0.301 + 0.263i)8-s + (0.725 + 0.633i)9-s + (0.00860 + 0.0635i)10-s + (0.537 − 0.321i)11-s + (−0.527 + 1.23i)12-s + (1.07 + 0.640i)13-s + (−0.101 + 0.0738i)14-s + (0.241 + 0.366i)15-s + (−0.872 − 0.0785i)16-s + (−0.353 + 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.271 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.271 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.346291 - 0.457454i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.346291 - 0.457454i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 + (-4.91 - 6.84i)T \) |
| good | 2 | \( 1 + (0.199 + 0.209i)T + (-0.0897 + 1.99i)T^{2} \) |
| 3 | \( 1 + (2.27 + 0.852i)T + (2.25 + 1.97i)T^{2} \) |
| 5 | \( 1 + (0.567 + 0.411i)T + (1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (-0.218 + 1.61i)T + (-6.74 - 1.86i)T^{2} \) |
| 11 | \( 1 + (-1.78 + 1.06i)T + (5.21 - 9.68i)T^{2} \) |
| 13 | \( 1 + (-3.86 - 2.30i)T + (6.16 + 11.4i)T^{2} \) |
| 17 | \( 1 + (1.45 - 4.48i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-4.05 + 6.13i)T + (-7.46 - 17.4i)T^{2} \) |
| 23 | \( 1 + (1.76 + 0.849i)T + (14.3 + 17.9i)T^{2} \) |
| 29 | \( 1 + (-8.22 + 2.26i)T + (24.8 - 14.8i)T^{2} \) |
| 31 | \( 1 + (2.23 - 0.200i)T + (30.5 - 5.53i)T^{2} \) |
| 37 | \( 1 + (0.704 - 0.339i)T + (23.0 - 28.9i)T^{2} \) |
| 41 | \( 1 + (1.81 - 7.97i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (-0.962 - 0.174i)T + (40.2 + 15.1i)T^{2} \) |
| 47 | \( 1 + (6.93 - 2.60i)T + (35.3 - 30.9i)T^{2} \) |
| 53 | \( 1 + (0.403 + 8.99i)T + (-52.7 + 4.75i)T^{2} \) |
| 59 | \( 1 + (3.08 - 7.22i)T + (-40.7 - 42.6i)T^{2} \) |
| 61 | \( 1 + (-1.91 - 14.1i)T + (-58.8 + 16.2i)T^{2} \) |
| 67 | \( 1 + (0.00938 - 0.208i)T + (-66.7 - 6.00i)T^{2} \) |
| 73 | \( 1 + (-4.60 - 4.82i)T + (-3.27 + 72.9i)T^{2} \) |
| 79 | \( 1 + (-1.04 + 0.911i)T + (10.6 - 78.2i)T^{2} \) |
| 83 | \( 1 + (-1.94 + 4.54i)T + (-57.3 - 59.9i)T^{2} \) |
| 89 | \( 1 + (0.723 + 16.1i)T + (-88.6 + 7.97i)T^{2} \) |
| 97 | \( 1 + (1.34 + 5.89i)T + (-87.3 + 42.0i)T^{2} \) |
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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
−14.21444232483499656681547129897, −13.25004604756751389349067419448, −11.74136970474370356649301411297, −11.22062449121556969616282969511, −10.20579279184253031831867572535, −8.674775135024072912619789006836, −6.75839278059251313785806972846, −6.03884866600217606764130968140, −4.53075372893221195673949293175, −1.07006760633836545428243130082,
3.61021847168725850569517794647, 5.27051863839743201055473435881, 6.56161727073164096016820877203, 7.985489229422641077713298165712, 9.370114113695403328803092778110, 10.80473812234268482770949809135, 11.81381368931514993740897177596, 12.30794568728037936827859552941, 13.85923534552888495748194394578, 15.59780043483875212064744788393
