| L(s) = 1 | + (0.707 + 1.88i)2-s + (−1.54 + 1.34i)4-s + (0.0743 + 0.102i)5-s + (2.17 − 1.43i)7-s + (−0.0898 − 0.0483i)8-s + (−0.140 + 0.212i)10-s + (0.582 − 0.608i)11-s + (0.602 − 0.575i)13-s + (4.24 + 3.08i)14-s + (−0.522 + 3.86i)16-s + (2.34 + 7.20i)17-s + (1.09 + 0.0985i)19-s + (−0.253 − 0.0577i)20-s + (1.55 + 0.666i)22-s + (−3.53 + 4.43i)23-s + ⋯ |
| L(s) = 1 | + (0.500 + 1.33i)2-s + (−0.772 + 0.674i)4-s + (0.0332 + 0.0457i)5-s + (0.823 − 0.543i)7-s + (−0.0317 − 0.0170i)8-s + (−0.0443 + 0.0672i)10-s + (0.175 − 0.183i)11-s + (0.166 − 0.159i)13-s + (1.13 + 0.825i)14-s + (−0.130 + 0.965i)16-s + (0.568 + 1.74i)17-s + (0.251 + 0.0226i)19-s + (−0.0565 − 0.0129i)20-s + (0.332 + 0.142i)22-s + (−0.737 + 0.924i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.282 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.282 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.31776 + 1.76089i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31776 + 1.76089i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 71 | \( 1 + (4.35 - 7.21i)T \) |
| good | 2 | \( 1 + (-0.707 - 1.88i)T + (-1.50 + 1.31i)T^{2} \) |
| 5 | \( 1 + (-0.0743 - 0.102i)T + (-1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (-2.17 + 1.43i)T + (2.75 - 6.43i)T^{2} \) |
| 11 | \( 1 + (-0.582 + 0.608i)T + (-0.493 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.602 + 0.575i)T + (0.583 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-2.34 - 7.20i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.09 - 0.0985i)T + (18.6 + 3.39i)T^{2} \) |
| 23 | \( 1 + (3.53 - 4.43i)T + (-5.11 - 22.4i)T^{2} \) |
| 29 | \( 1 + (-8.03 + 3.43i)T + (20.0 - 20.9i)T^{2} \) |
| 31 | \( 1 + (2.67 - 0.361i)T + (29.8 - 8.24i)T^{2} \) |
| 37 | \( 1 + (3.61 + 4.52i)T + (-8.23 + 36.0i)T^{2} \) |
| 41 | \( 1 + (3.36 - 1.61i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (5.58 + 1.54i)T + (36.9 + 22.0i)T^{2} \) |
| 47 | \( 1 + (-5.33 + 3.18i)T + (22.2 - 41.3i)T^{2} \) |
| 53 | \( 1 + (5.31 + 4.63i)T + (7.11 + 52.5i)T^{2} \) |
| 59 | \( 1 + (0.0224 - 0.00408i)T + (55.2 - 20.7i)T^{2} \) |
| 61 | \( 1 + (-0.862 - 0.569i)T + (23.9 + 56.0i)T^{2} \) |
| 67 | \( 1 + (-3.69 - 4.22i)T + (-8.99 + 66.3i)T^{2} \) |
| 73 | \( 1 + (-0.604 + 0.226i)T + (54.9 - 48.0i)T^{2} \) |
| 79 | \( 1 + (-2.83 + 5.26i)T + (-43.5 - 65.9i)T^{2} \) |
| 83 | \( 1 + (0.681 + 3.75i)T + (-77.7 + 29.1i)T^{2} \) |
| 89 | \( 1 + (-2.15 + 2.46i)T + (-11.9 - 88.1i)T^{2} \) |
| 97 | \( 1 + (-7.12 + 14.7i)T + (-60.4 - 75.8i)T^{2} \) |
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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
−10.70624168213065026182878890410, −10.07717612095804102021759044148, −8.499034477838010369635976178903, −8.129079361323657525044047240177, −7.23937703866175532888744243650, −6.28043154710544224708278715332, −5.56951468941570019537397934108, −4.52855247174376189120213309275, −3.68115680727960380033650159489, −1.64693195549937758348163304847,
1.25320657499247061696550443978, 2.44954201353350244582736429580, 3.40494782854190977844860973587, 4.73784458104343739938331204458, 5.19349725714285076272527328675, 6.72749369856796011069483724205, 7.76725361094781362057502840667, 8.856757773331847971558036557312, 9.683999462502041528953596477841, 10.54406359815355762284314618952
