| L(s) = 1 | + (−0.565 − 0.807i)2-s + (1.55 + 0.756i)3-s + (0.351 − 0.966i)4-s + (−0.542 − 2.16i)5-s + (−0.270 − 1.68i)6-s + (−0.590 − 0.275i)7-s + (−2.88 + 0.772i)8-s + (1.85 + 2.35i)9-s + (−1.44 + 1.66i)10-s + (0.890 − 1.06i)11-s + (1.27 − 1.24i)12-s + (4.19 + 2.93i)13-s + (0.111 + 0.632i)14-s + (0.794 − 3.79i)15-s + (0.677 + 0.568i)16-s + (−4.41 − 1.18i)17-s + ⋯ |
| L(s) = 1 | + (−0.399 − 0.570i)2-s + (0.899 + 0.436i)3-s + (0.175 − 0.483i)4-s + (−0.242 − 0.970i)5-s + (−0.110 − 0.688i)6-s + (−0.223 − 0.104i)7-s + (−1.01 + 0.273i)8-s + (0.618 + 0.785i)9-s + (−0.456 + 0.526i)10-s + (0.268 − 0.319i)11-s + (0.369 − 0.357i)12-s + (1.16 + 0.814i)13-s + (0.0298 + 0.169i)14-s + (0.205 − 0.978i)15-s + (0.169 + 0.142i)16-s + (−1.06 − 0.286i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.418 + 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.418 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.968003 - 0.619609i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.968003 - 0.619609i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.55 - 0.756i)T \) |
| 5 | \( 1 + (0.542 + 2.16i)T \) |
| good | 2 | \( 1 + (0.565 + 0.807i)T + (-0.684 + 1.87i)T^{2} \) |
| 7 | \( 1 + (0.590 + 0.275i)T + (4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (-0.890 + 1.06i)T + (-1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-4.19 - 2.93i)T + (4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (4.41 + 1.18i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (0.00652 + 0.00376i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.12 - 6.70i)T + (-14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (-0.586 + 3.32i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-3.73 - 1.35i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (0.430 - 1.60i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (1.90 - 0.335i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (10.6 - 0.929i)T + (42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (3.08 - 6.61i)T + (-30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (-1.18 - 1.18i)T + 53iT^{2} \) |
| 59 | \( 1 + (-7.77 + 6.52i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (9.02 - 3.28i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-8.62 + 12.3i)T + (-22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (-5.41 + 3.12i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (3.25 + 12.1i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-0.782 - 0.137i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-7.59 + 5.32i)T + (28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (5.89 - 10.2i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.190 + 2.18i)T + (-95.5 + 16.8i)T^{2} \) |
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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
−13.26962406443821637779685002908, −11.74233325598502358921726036860, −11.00323443400905037945610913814, −9.669114634260985547234117892832, −9.080917614173144369526507966623, −8.246891331366827183518770277387, −6.50887937560392824879348517729, −4.88804527672734923621264800977, −3.48430736993754955409640923203, −1.63068859849867891493788605589,
2.68815475510056706041409599424, 3.77423867323507996560031698935, 6.43703270753709234003912274473, 6.93279568574511960037008693090, 8.192434577951354017483089391694, 8.767015689055095101127156842442, 10.17683518335111016790732684027, 11.40698405426316285057978803374, 12.61770823337817574280096731315, 13.39799067843927879831570582459
