| 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
−13.39799067843927879831570582459, −12.61770823337817574280096731315, −11.40698405426316285057978803374, −10.17683518335111016790732684027, −8.767015689055095101127156842442, −8.192434577951354017483089391694, −6.93279568574511960037008693090, −6.43703270753709234003912274473, −3.77423867323507996560031698935, −2.68815475510056706041409599424,
1.63068859849867891493788605589, 3.48430736993754955409640923203, 4.88804527672734923621264800977, 6.50887937560392824879348517729, 8.246891331366827183518770277387, 9.080917614173144369526507966623, 9.669114634260985547234117892832, 11.00323443400905037945610913814, 11.74233325598502358921726036860, 13.26962406443821637779685002908
