| L(s) = 1 | + (−1.71 − 2.36i)2-s + (−2.67 + 1.35i)3-s + (−1.40 + 4.32i)4-s + (2.09 + 4.54i)5-s + (7.79 + 4.01i)6-s + 3.73·7-s + (1.50 − 0.490i)8-s + (5.34 − 7.23i)9-s + (7.13 − 12.7i)10-s + (10.5 + 14.5i)11-s + (−2.07 − 13.4i)12-s + (−1.50 − 1.09i)13-s + (−6.41 − 8.83i)14-s + (−11.7 − 9.33i)15-s + (10.9 + 7.95i)16-s + (−7.75 + 2.52i)17-s + ⋯ |
| L(s) = 1 | + (−0.858 − 1.18i)2-s + (−0.892 + 0.450i)3-s + (−0.350 + 1.08i)4-s + (0.418 + 0.908i)5-s + (1.29 + 0.668i)6-s + 0.533·7-s + (0.188 − 0.0612i)8-s + (0.594 − 0.804i)9-s + (0.713 − 1.27i)10-s + (0.961 + 1.32i)11-s + (−0.173 − 1.12i)12-s + (−0.115 − 0.0840i)13-s + (−0.458 − 0.631i)14-s + (−0.782 − 0.622i)15-s + (0.684 + 0.497i)16-s + (−0.456 + 0.148i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.199i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.979 - 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.645174 + 0.0649256i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.645174 + 0.0649256i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.67 - 1.35i)T \) |
| 5 | \( 1 + (-2.09 - 4.54i)T \) |
| good | 2 | \( 1 + (1.71 + 2.36i)T + (-1.23 + 3.80i)T^{2} \) |
| 7 | \( 1 - 3.73T + 49T^{2} \) |
| 11 | \( 1 + (-10.5 - 14.5i)T + (-37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (1.50 + 1.09i)T + (52.2 + 160. i)T^{2} \) |
| 17 | \( 1 + (7.75 - 2.52i)T + (233. - 169. i)T^{2} \) |
| 19 | \( 1 + (-7.94 - 24.4i)T + (-292. + 212. i)T^{2} \) |
| 23 | \( 1 + (-9.59 - 13.2i)T + (-163. + 503. i)T^{2} \) |
| 29 | \( 1 + (18.4 + 5.99i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (15.1 + 46.5i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (-1.93 - 1.40i)T + (423. + 1.30e3i)T^{2} \) |
| 41 | \( 1 + (-22.4 + 30.8i)T + (-519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 - 2.68T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-76.6 - 24.9i)T + (1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (28.2 + 9.16i)T + (2.27e3 + 1.65e3i)T^{2} \) |
| 59 | \( 1 + (-13.8 + 19.1i)T + (-1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-37.9 + 27.5i)T + (1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 + (-21.8 - 67.3i)T + (-3.63e3 + 2.63e3i)T^{2} \) |
| 71 | \( 1 + (106. + 34.5i)T + (4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-67.2 + 48.8i)T + (1.64e3 - 5.06e3i)T^{2} \) |
| 79 | \( 1 + (31.0 - 95.5i)T + (-5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-21.2 + 6.92i)T + (5.57e3 - 4.04e3i)T^{2} \) |
| 89 | \( 1 + (-19.6 - 27.0i)T + (-2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-39.1 + 120. i)T + (-7.61e3 - 5.53e3i)T^{2} \) |
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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
−14.50527684974338837114007579585, −12.66173216827293874214848815195, −11.66868715958782200785471538496, −10.99736177251616973644045721519, −9.956565961510225693877010824140, −9.376019065890188066464466652336, −7.38783779022285499588426902501, −5.87581200089689135035458978593, −3.92295120271341285771516363069, −1.81193384229524481603100449933,
0.897213420566254642961809842833, 4.99144705357412177188947261631, 6.07849023587609760046545875249, 7.12954924510144044388403589742, 8.510315028808479179382029377218, 9.243358835109606455612896373939, 10.91580023092283517377040746360, 12.00077147876196042713800764037, 13.28642330209829675147373663956, 14.38193125121654493113411626842
