| L(s) = 1 | + (−1.46 + 1.36i)2-s + (9.21 − 1.38i)3-s + (0.298 − 3.98i)4-s + (2.03 − 5.19i)5-s + (−11.6 + 14.5i)6-s + (8.54 − 16.4i)7-s + (4.98 + 6.25i)8-s + (57.1 − 17.6i)9-s + (4.07 + 10.3i)10-s + (−62.5 − 19.2i)11-s + (−2.78 − 37.1i)12-s + (2.99 + 13.1i)13-s + (9.82 + 35.7i)14-s + (11.5 − 50.6i)15-s + (−15.8 − 2.38i)16-s + (−10.0 + 6.85i)17-s + ⋯ |
| L(s) = 1 | + (−0.518 + 0.480i)2-s + (1.77 − 0.267i)3-s + (0.0373 − 0.498i)4-s + (0.182 − 0.464i)5-s + (−0.790 + 0.990i)6-s + (0.461 − 0.887i)7-s + (0.220 + 0.276i)8-s + (2.11 − 0.652i)9-s + (0.128 + 0.328i)10-s + (−1.71 − 0.528i)11-s + (−0.0669 − 0.893i)12-s + (0.0638 + 0.279i)13-s + (0.187 + 0.681i)14-s + (0.199 − 0.872i)15-s + (−0.247 − 0.0372i)16-s + (−0.143 + 0.0978i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.312i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.949 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.09949 - 0.336704i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.09949 - 0.336704i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.46 - 1.36i)T \) |
| 7 | \( 1 + (-8.54 + 16.4i)T \) |
| good | 3 | \( 1 + (-9.21 + 1.38i)T + (25.8 - 7.95i)T^{2} \) |
| 5 | \( 1 + (-2.03 + 5.19i)T + (-91.6 - 85.0i)T^{2} \) |
| 11 | \( 1 + (62.5 + 19.2i)T + (1.09e3 + 749. i)T^{2} \) |
| 13 | \( 1 + (-2.99 - 13.1i)T + (-1.97e3 + 953. i)T^{2} \) |
| 17 | \( 1 + (10.0 - 6.85i)T + (1.79e3 - 4.57e3i)T^{2} \) |
| 19 | \( 1 + (-39.4 - 68.2i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-136. - 93.1i)T + (4.44e3 + 1.13e4i)T^{2} \) |
| 29 | \( 1 + (23.3 + 11.2i)T + (1.52e4 + 1.90e4i)T^{2} \) |
| 31 | \( 1 + (83.7 - 145. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-19.3 - 258. i)T + (-5.00e4 + 7.54e3i)T^{2} \) |
| 41 | \( 1 + (176. + 221. i)T + (-1.53e4 + 6.71e4i)T^{2} \) |
| 43 | \( 1 + (-149. + 187. i)T + (-1.76e4 - 7.75e4i)T^{2} \) |
| 47 | \( 1 + (399. - 371. i)T + (7.75e3 - 1.03e5i)T^{2} \) |
| 53 | \( 1 + (6.51 - 86.9i)T + (-1.47e5 - 2.21e4i)T^{2} \) |
| 59 | \( 1 + (15.6 + 39.9i)T + (-1.50e5 + 1.39e5i)T^{2} \) |
| 61 | \( 1 + (49.2 + 656. i)T + (-2.24e5 + 3.38e4i)T^{2} \) |
| 67 | \( 1 + (-148. + 257. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (801. - 385. i)T + (2.23e5 - 2.79e5i)T^{2} \) |
| 73 | \( 1 + (-752. - 698. i)T + (2.90e4 + 3.87e5i)T^{2} \) |
| 79 | \( 1 + (402. + 696. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-100. + 440. i)T + (-5.15e5 - 2.48e5i)T^{2} \) |
| 89 | \( 1 + (358. - 110. i)T + (5.82e5 - 3.97e5i)T^{2} \) |
| 97 | \( 1 - 1.21e3T + 9.12e5T^{2} \) |
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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
−13.56737503032557901839447331542, −12.91641242256992418448660939494, −10.84995122235462904018349877034, −9.785217234790323484916005148177, −8.708104405975278706685309508037, −7.931738589902094369265001988095, −7.17022112795982205058894331791, −5.05973175940739040476204635925, −3.24915303861711033201902679510, −1.47612302447516385156930739166,
2.32312687067067518352870179772, 2.92897934807548018104085750140, 4.86198323641949473451849131770, 7.28918688721685707069882553586, 8.212895798224834079818934737406, 9.055604018970682249713652054527, 10.04588966242650455855917471960, 11.01840343717750482279791957067, 12.73717474164505958146439582769, 13.37814557715485282169963590941
