| L(s) = 1 | + (0.302 − 0.219i)2-s + (−0.0950 + 0.904i)3-s + (−0.574 + 1.76i)4-s + (−0.5 + 0.866i)5-s + (0.169 + 0.294i)6-s + (−3.29 + 0.701i)7-s + (0.445 + 1.37i)8-s + (2.12 + 0.451i)9-s + (0.0390 + 0.371i)10-s + (0.241 + 0.268i)11-s + (−1.54 − 0.688i)12-s + (1.17 − 0.524i)13-s + (−0.842 + 0.935i)14-s + (−0.735 − 0.534i)15-s + (−2.57 − 1.87i)16-s + (3.02 − 3.36i)17-s + ⋯ |
| L(s) = 1 | + (0.213 − 0.155i)2-s + (−0.0548 + 0.522i)3-s + (−0.287 + 0.884i)4-s + (−0.223 + 0.387i)5-s + (0.0693 + 0.120i)6-s + (−1.24 + 0.265i)7-s + (0.157 + 0.484i)8-s + (0.708 + 0.150i)9-s + (0.0123 + 0.117i)10-s + (0.0727 + 0.0808i)11-s + (−0.446 − 0.198i)12-s + (0.326 − 0.145i)13-s + (−0.225 + 0.250i)14-s + (−0.189 − 0.138i)15-s + (−0.643 − 0.467i)16-s + (0.734 − 0.815i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0584 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0584 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.759393 + 0.716211i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.759393 + 0.716211i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (-0.329 + 5.55i)T \) |
| good | 2 | \( 1 + (-0.302 + 0.219i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (0.0950 - 0.904i)T + (-2.93 - 0.623i)T^{2} \) |
| 7 | \( 1 + (3.29 - 0.701i)T + (6.39 - 2.84i)T^{2} \) |
| 11 | \( 1 + (-0.241 - 0.268i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-1.17 + 0.524i)T + (8.69 - 9.66i)T^{2} \) |
| 17 | \( 1 + (-3.02 + 3.36i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-5.53 - 2.46i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (-1.31 - 4.05i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-3.32 + 2.41i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (5.04 + 8.74i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.14 - 10.8i)T + (-40.1 + 8.52i)T^{2} \) |
| 43 | \( 1 + (4.09 + 1.82i)T + (28.7 + 31.9i)T^{2} \) |
| 47 | \( 1 + (3.72 + 2.70i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-3.04 - 0.647i)T + (48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (0.396 - 3.77i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 + (-3.48 + 6.04i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.02 + 0.643i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (0.672 + 0.746i)T + (-7.63 + 72.6i)T^{2} \) |
| 79 | \( 1 + (-2.08 + 2.31i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-1.82 - 17.3i)T + (-81.1 + 17.2i)T^{2} \) |
| 89 | \( 1 + (0.516 - 1.58i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (0.695 - 2.14i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−13.13794152792398097648249951810, −12.23226011211826302615836131280, −11.37480341277861128097518309396, −9.937874549787969894434883958185, −9.412245569280581474844369577884, −7.891052817286673516057405254595, −6.96240023546993226031747840523, −5.38187446995231389568427992066, −3.88164394782887516688661740419, −3.06085789378574804642488814624,
1.09451623426906677896337627658, 3.59628943483545288892930323769, 5.04320854943091749111370993001, 6.37389728436481984932241400676, 7.06536661383331293796066152834, 8.661871164576275917709013339813, 9.791181370507143469820268781025, 10.44149610625163438321944894968, 12.00811379747187789463207442128, 12.89320909908321225114284710329
