| L(s) = 1 | + (−0.390 − 0.920i)2-s + (−2.94 + 0.571i)3-s + (−0.694 + 0.719i)4-s + (1.38 + 1.75i)5-s + (1.67 + 2.48i)6-s + (−1.10 + 4.14i)7-s + (0.933 + 0.358i)8-s + (5.54 − 2.23i)9-s + (1.07 − 1.96i)10-s + (0.841 + 0.178i)11-s + (1.63 − 2.51i)12-s + (4.42 − 0.543i)13-s + (4.24 − 0.596i)14-s + (−5.08 − 4.35i)15-s + (−0.0348 − 0.999i)16-s + (−0.0586 + 0.0976i)17-s + ⋯ |
| L(s) = 1 | + (−0.276 − 0.650i)2-s + (−1.69 + 0.330i)3-s + (−0.347 + 0.359i)4-s + (0.621 + 0.783i)5-s + (0.684 + 1.01i)6-s + (−0.419 + 1.56i)7-s + (0.330 + 0.126i)8-s + (1.84 − 0.746i)9-s + (0.338 − 0.620i)10-s + (0.253 + 0.0539i)11-s + (0.471 − 0.725i)12-s + (1.22 − 0.150i)13-s + (1.13 − 0.159i)14-s + (−1.31 − 1.12i)15-s + (−0.00872 − 0.249i)16-s + (−0.0142 + 0.0236i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.501748 + 0.575248i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.501748 + 0.575248i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.390 + 0.920i)T \) |
| 5 | \( 1 + (-1.38 - 1.75i)T \) |
| 19 | \( 1 + (-4.23 - 1.02i)T \) |
| good | 3 | \( 1 + (2.94 - 0.571i)T + (2.78 - 1.12i)T^{2} \) |
| 7 | \( 1 + (1.10 - 4.14i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-0.841 - 0.178i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-4.42 + 0.543i)T + (12.6 - 3.14i)T^{2} \) |
| 17 | \( 1 + (0.0586 - 0.0976i)T + (-7.98 - 15.0i)T^{2} \) |
| 23 | \( 1 + (1.53 - 5.03i)T + (-19.0 - 12.8i)T^{2} \) |
| 29 | \( 1 + (-0.513 + 2.06i)T + (-25.6 - 13.6i)T^{2} \) |
| 31 | \( 1 + (-1.23 - 0.129i)T + (30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (2.64 + 5.19i)T + (-21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (-5.05 + 0.176i)T + (40.9 - 2.86i)T^{2} \) |
| 43 | \( 1 + (-5.46 - 7.79i)T + (-14.7 + 40.4i)T^{2} \) |
| 47 | \( 1 + (-3.71 + 2.23i)T + (22.0 - 41.4i)T^{2} \) |
| 53 | \( 1 + (-0.164 - 9.40i)T + (-52.9 + 1.84i)T^{2} \) |
| 59 | \( 1 + (6.93 - 4.33i)T + (25.8 - 53.0i)T^{2} \) |
| 61 | \( 1 + (-1.16 + 0.621i)T + (34.1 - 50.5i)T^{2} \) |
| 67 | \( 1 + (1.03 - 1.19i)T + (-9.32 - 66.3i)T^{2} \) |
| 71 | \( 1 + (0.182 - 0.0890i)T + (43.7 - 55.9i)T^{2} \) |
| 73 | \( 1 + (3.74 + 0.459i)T + (70.8 + 17.6i)T^{2} \) |
| 79 | \( 1 + (-8.35 + 12.3i)T + (-29.5 - 73.2i)T^{2} \) |
| 83 | \( 1 + (-0.749 - 0.925i)T + (-17.2 + 81.1i)T^{2} \) |
| 89 | \( 1 + (0.593 - 16.9i)T + (-88.7 - 6.20i)T^{2} \) |
| 97 | \( 1 + (-10.1 + 8.85i)T + (13.4 - 96.0i)T^{2} \) |
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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
−10.50751494399778492967493826108, −9.499383094789486355232880442477, −9.177333926446237032015065405442, −7.68268609574848622242231528909, −6.45147913734874860495141276255, −5.84382707120576572722876711725, −5.39704963493283104727452192734, −3.94704752209219379746774333253, −2.80940118323903852363833906861, −1.39642257836942346149900513446,
0.60659778419633160610473609586, 1.30318240323069685867872547190, 3.95688615752700454885674880109, 4.78754050305189898697010944551, 5.67097849568726260797700902468, 6.44164979540782321157667991016, 6.91766239192061816819245369347, 7.922510557030432373182325292893, 9.033595247528103935092956888273, 10.00907942594206251611952107971
