| L(s) = 1 | + (0.445 − 1.94i)2-s + (−1.23 + 1.55i)3-s + (−3.60 − 1.73i)4-s + (−7.44 + 9.33i)5-s + (2.47 + 3.10i)6-s + (11.3 − 14.5i)7-s + (−4.98 + 6.25i)8-s + (5.12 + 22.4i)9-s + (14.8 + 18.6i)10-s + (−13.0 + 57.1i)11-s + (7.16 − 3.44i)12-s + (−13.3 + 58.2i)13-s + (−23.3 − 28.7i)14-s + (−5.27 − 23.1i)15-s + (9.97 + 12.5i)16-s + (44.8 − 21.6i)17-s + ⋯ |
| L(s) = 1 | + (0.157 − 0.689i)2-s + (−0.238 + 0.298i)3-s + (−0.450 − 0.216i)4-s + (−0.665 + 0.834i)5-s + (0.168 + 0.211i)6-s + (0.615 − 0.788i)7-s + (−0.220 + 0.276i)8-s + (0.189 + 0.832i)9-s + (0.470 + 0.590i)10-s + (−0.357 + 1.56i)11-s + (0.172 − 0.0829i)12-s + (−0.283 + 1.24i)13-s + (−0.446 − 0.548i)14-s + (−0.0908 − 0.398i)15-s + (0.155 + 0.195i)16-s + (0.640 − 0.308i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.907929 + 0.604217i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.907929 + 0.604217i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.445 + 1.94i)T \) |
| 7 | \( 1 + (-11.3 + 14.5i)T \) |
| good | 3 | \( 1 + (1.23 - 1.55i)T + (-6.00 - 26.3i)T^{2} \) |
| 5 | \( 1 + (7.44 - 9.33i)T + (-27.8 - 121. i)T^{2} \) |
| 11 | \( 1 + (13.0 - 57.1i)T + (-1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (13.3 - 58.2i)T + (-1.97e3 - 953. i)T^{2} \) |
| 17 | \( 1 + (-44.8 + 21.6i)T + (3.06e3 - 3.84e3i)T^{2} \) |
| 19 | \( 1 - 51.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + (188. + 90.9i)T + (7.58e3 + 9.51e3i)T^{2} \) |
| 29 | \( 1 + (143. - 69.2i)T + (1.52e4 - 1.90e4i)T^{2} \) |
| 31 | \( 1 - 309.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (174. - 84.1i)T + (3.15e4 - 3.96e4i)T^{2} \) |
| 41 | \( 1 + (-166. + 209. i)T + (-1.53e4 - 6.71e4i)T^{2} \) |
| 43 | \( 1 + (176. + 221. i)T + (-1.76e4 + 7.75e4i)T^{2} \) |
| 47 | \( 1 + (87.4 - 383. i)T + (-9.35e4 - 4.50e4i)T^{2} \) |
| 53 | \( 1 + (56.3 + 27.1i)T + (9.28e4 + 1.16e5i)T^{2} \) |
| 59 | \( 1 + (-459. - 576. i)T + (-4.57e4 + 2.00e5i)T^{2} \) |
| 61 | \( 1 + (151. - 73.0i)T + (1.41e5 - 1.77e5i)T^{2} \) |
| 67 | \( 1 - 990.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-480. - 231. i)T + (2.23e5 + 2.79e5i)T^{2} \) |
| 73 | \( 1 + (188. + 827. i)T + (-3.50e5 + 1.68e5i)T^{2} \) |
| 79 | \( 1 - 535.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-93.9 - 411. i)T + (-5.15e5 + 2.48e5i)T^{2} \) |
| 89 | \( 1 + (-90.5 - 396. i)T + (-6.35e5 + 3.05e5i)T^{2} \) |
| 97 | \( 1 - 487.T + 9.12e5T^{2} \) |
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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
−13.79190127518903334298447981381, −12.24145509950974150864148216958, −11.48915313852434297370357417410, −10.45582343818760735910637124421, −9.837624265025950520624986598513, −7.88277948319749712977180277585, −7.05042839219463074557003667011, −4.89765009874717647867493729764, −4.02876215121861621348149023962, −2.07786794541886195170731315528,
0.63064640478901586467910954992, 3.54520153664428804135276982772, 5.27068582990738742150112714889, 6.04552181008660799834916770552, 7.998372401220408445924523125775, 8.259988280998053379421635684914, 9.750398532090308722361534665343, 11.54060969702162102313056921661, 12.20205556368899045550928468523, 13.18418290064916518161482948179
