| L(s) = 1 | + (−0.104 + 0.994i)2-s + (0.809 + 0.587i)3-s + (−0.978 − 0.207i)4-s + (2.23 − 2.48i)5-s + (−0.669 + 0.743i)6-s + (0.0234 + 0.0104i)7-s + (0.309 − 0.951i)8-s + (0.309 + 0.951i)9-s + (2.23 + 2.48i)10-s + 2.03·11-s + (−0.669 − 0.743i)12-s + (−1.07 − 1.86i)13-s + (−0.0128 + 0.0222i)14-s + (3.27 − 0.695i)15-s + (0.913 + 0.406i)16-s + (0.146 + 0.0310i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 + 0.703i)2-s + (0.467 + 0.339i)3-s + (−0.489 − 0.103i)4-s + (1.00 − 1.11i)5-s + (−0.273 + 0.303i)6-s + (0.00885 + 0.00394i)7-s + (0.109 − 0.336i)8-s + (0.103 + 0.317i)9-s + (0.708 + 0.786i)10-s + 0.612·11-s + (−0.193 − 0.214i)12-s + (−0.297 − 0.515i)13-s + (−0.00342 + 0.00593i)14-s + (0.845 − 0.179i)15-s + (0.228 + 0.101i)16-s + (0.0354 + 0.00754i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.505i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.862 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.64406 + 0.446584i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64406 + 0.446584i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.104 - 0.994i)T \) |
| 3 | \( 1 + (-0.809 - 0.587i)T \) |
| 61 | \( 1 + (-5.74 + 5.29i)T \) |
| good | 5 | \( 1 + (-2.23 + 2.48i)T + (-0.522 - 4.97i)T^{2} \) |
| 7 | \( 1 + (-0.0234 - 0.0104i)T + (4.68 + 5.20i)T^{2} \) |
| 11 | \( 1 - 2.03T + 11T^{2} \) |
| 13 | \( 1 + (1.07 + 1.86i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.146 - 0.0310i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (-5.19 + 2.31i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-0.999 - 3.07i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (5.09 - 8.82i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.217 + 2.06i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (5.06 - 3.67i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (6.25 - 4.54i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-2.29 + 0.488i)T + (39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (2.64 - 4.57i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.423 + 1.30i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-0.762 + 7.25i)T + (-57.7 - 12.2i)T^{2} \) |
| 67 | \( 1 + (0.0360 - 0.0400i)T + (-7.00 - 66.6i)T^{2} \) |
| 71 | \( 1 + (-4.81 - 5.35i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (8.45 + 9.38i)T + (-7.63 + 72.6i)T^{2} \) |
| 79 | \( 1 + (3.16 - 0.673i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (-0.189 + 1.80i)T + (-81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (1.47 + 1.07i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (0.386 + 3.68i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−11.51720069322471679203509967025, −10.12454846591684038028120434434, −9.399454293258600952718828486151, −8.893770048423370271327185596492, −7.86050698657516831952112658350, −6.73831401802124166557682403140, −5.40588937619278919296731278905, −4.95883374134814166134935216789, −3.40120435386124526056230049562, −1.48083681670083325327393519056,
1.74463120661075674832189865390, 2.76375428547576889094749725812, 3.90060466279429175609632394134, 5.56326218048163428385965090671, 6.63992964264375496606299759273, 7.51639694672859193703509698522, 8.839825435561582858373084007105, 9.690382331793438811733605605154, 10.27175737015357921344130603884, 11.36034824172670422902432348012
