| L(s) = 1 | + (1.26 − 0.640i)2-s + (1.37 − 1.05i)3-s + (1.18 − 1.61i)4-s + (0.347 − 0.396i)5-s + (1.05 − 2.20i)6-s + (2.56 − 0.338i)7-s + (0.454 − 2.79i)8-s + (0.773 − 2.89i)9-s + (0.184 − 0.723i)10-s + (−1.93 + 3.93i)11-s + (−0.0829 − 3.46i)12-s + (−1.46 + 4.31i)13-s + (3.02 − 2.07i)14-s + (0.0592 − 0.912i)15-s + (−1.21 − 3.81i)16-s + (−3.42 + 3.42i)17-s + ⋯ |
| L(s) = 1 | + (0.891 − 0.452i)2-s + (0.793 − 0.609i)3-s + (0.590 − 0.807i)4-s + (0.155 − 0.177i)5-s + (0.431 − 0.902i)6-s + (0.970 − 0.127i)7-s + (0.160 − 0.987i)8-s + (0.257 − 0.966i)9-s + (0.0584 − 0.228i)10-s + (−0.584 + 1.18i)11-s + (−0.0239 − 0.999i)12-s + (−0.406 + 1.19i)13-s + (0.807 − 0.553i)14-s + (0.0153 − 0.235i)15-s + (−0.303 − 0.952i)16-s + (−0.831 + 0.831i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.55813 - 2.06370i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.55813 - 2.06370i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.26 + 0.640i)T \) |
| 3 | \( 1 + (-1.37 + 1.05i)T \) |
| good | 5 | \( 1 + (-0.347 + 0.396i)T + (-0.652 - 4.95i)T^{2} \) |
| 7 | \( 1 + (-2.56 + 0.338i)T + (6.76 - 1.81i)T^{2} \) |
| 11 | \( 1 + (1.93 - 3.93i)T + (-6.69 - 8.72i)T^{2} \) |
| 13 | \( 1 + (1.46 - 4.31i)T + (-10.3 - 7.91i)T^{2} \) |
| 17 | \( 1 + (3.42 - 3.42i)T - 17iT^{2} \) |
| 19 | \( 1 + (0.675 + 3.39i)T + (-17.5 + 7.27i)T^{2} \) |
| 23 | \( 1 + (0.716 - 5.44i)T + (-22.2 - 5.95i)T^{2} \) |
| 29 | \( 1 + (2.67 + 0.175i)T + (28.7 + 3.78i)T^{2} \) |
| 31 | \( 1 + (3.23 + 1.86i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.61 + 8.12i)T + (-34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (-1.30 + 9.91i)T + (-39.6 - 10.6i)T^{2} \) |
| 43 | \( 1 + (-4.70 - 2.32i)T + (26.1 + 34.1i)T^{2} \) |
| 47 | \( 1 + (-6.57 - 1.76i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.45 - 3.67i)T + (-20.2 + 48.9i)T^{2} \) |
| 59 | \( 1 + (5.61 - 6.40i)T + (-7.70 - 58.4i)T^{2} \) |
| 61 | \( 1 + (-0.992 + 15.1i)T + (-60.4 - 7.96i)T^{2} \) |
| 67 | \( 1 + (9.00 - 4.44i)T + (40.7 - 53.1i)T^{2} \) |
| 71 | \( 1 + (-3.11 + 7.52i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (-4.52 - 10.9i)T + (-51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (-0.461 + 1.72i)T + (-68.4 - 39.5i)T^{2} \) |
| 83 | \( 1 + (10.8 - 9.49i)T + (10.8 - 82.2i)T^{2} \) |
| 89 | \( 1 + (1.48 + 0.616i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + (-15.5 + 9.00i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.82323429265854713110885465421, −9.543885373914627042837626970804, −8.989429760957924153442350833756, −7.45188055520073715715346140722, −7.18562034435270113594758250460, −5.81333323837263794845491059129, −4.65852200568847761286338770321, −3.91460271390949273242319071063, −2.25096501287024878698213526278, −1.75020544229334320525919893697,
2.38762414996017348394184266489, 3.15515192365973845473219132848, 4.45431233358642520834850509662, 5.17417602316979721621680974645, 6.14742785290094009145841734273, 7.54995061554631899788983678906, 8.179244575248380522042532353548, 8.785373447243140567723128303445, 10.29299152593614599231839654326, 10.85574620988969062910163864202
