L(s) = 1 | + (−0.939 + 0.342i)2-s + (2.77 + 1.00i)3-s + (0.766 − 0.642i)4-s + (−0.173 − 0.984i)5-s − 2.95·6-s + (−0.530 − 3.00i)7-s + (−0.500 + 0.866i)8-s + (4.37 + 3.66i)9-s + (0.5 + 0.866i)10-s + (1.06 − 1.83i)11-s + (2.77 − 1.00i)12-s + (0.822 − 0.690i)13-s + (1.52 + 2.64i)14-s + (0.512 − 2.90i)15-s + (0.173 − 0.984i)16-s + (4.15 + 3.48i)17-s + ⋯ |
L(s) = 1 | + (−0.664 + 0.241i)2-s + (1.60 + 0.582i)3-s + (0.383 − 0.321i)4-s + (−0.0776 − 0.440i)5-s − 1.20·6-s + (−0.200 − 1.13i)7-s + (−0.176 + 0.306i)8-s + (1.45 + 1.22i)9-s + (0.158 + 0.273i)10-s + (0.319 − 0.553i)11-s + (0.800 − 0.291i)12-s + (0.228 − 0.191i)13-s + (0.408 + 0.707i)14-s + (0.132 − 0.750i)15-s + (0.0434 − 0.246i)16-s + (1.00 + 0.846i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.148i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.65743 + 0.123479i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.65743 + 0.123479i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.939 - 0.342i)T \) |
| 5 | \( 1 + (0.173 + 0.984i)T \) |
| 37 | \( 1 + (5.36 + 2.87i)T \) |
good | 3 | \( 1 + (-2.77 - 1.00i)T + (2.29 + 1.92i)T^{2} \) |
| 7 | \( 1 + (0.530 + 3.00i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-1.06 + 1.83i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.822 + 0.690i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-4.15 - 3.48i)T + (2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (-1.64 - 0.600i)T + (14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (0.696 + 1.20i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (5.11 - 8.86i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 41 | \( 1 + (-0.111 + 0.0934i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 - 7.30T + 43T^{2} \) |
| 47 | \( 1 + (-3.70 - 6.41i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.782 + 4.43i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (1.29 - 7.34i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-1.41 + 1.18i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (1.62 + 9.24i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (6.84 + 2.49i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + 15.7T + 73T^{2} \) |
| 79 | \( 1 + (-2.01 - 11.4i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-2.29 - 1.92i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-0.129 + 0.735i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (4.30 + 7.45i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.86202240350494838124774662709, −10.34060725658976330763333458286, −9.292602946764414476437968166059, −8.803958242521866012067639852505, −7.80952375068936195636536060122, −7.23402710502511267311417189225, −5.59161573023952369250626058978, −3.98890950041495298250886590602, −3.33106077192939322351058609714, −1.49839822815320827699035229058,
1.84565244230539308718711008582, 2.75645493917272060974379519903, 3.74315998461650312200722350954, 5.79864128465726379904429666271, 7.21126440135074485450939092713, 7.63172238831679733075050273694, 8.803891855031373393588006695045, 9.285538976697980204111292089637, 10.03957577233358143155867615145, 11.54181929178530331617151311271