| L(s) = 1 | + (−0.104 − 0.994i)2-s + (1.67 + 0.438i)3-s + (−0.978 + 0.207i)4-s + (−0.684 − 1.53i)5-s + (0.261 − 1.71i)6-s + (1.54 + 2.14i)7-s + (0.309 + 0.951i)8-s + (2.61 + 1.47i)9-s + (−1.45 + 0.841i)10-s + (2.83 + 1.72i)11-s + (−1.73 − 0.0808i)12-s + (1.78 + 2.46i)13-s + (1.97 − 1.76i)14-s + (−0.472 − 2.87i)15-s + (0.913 − 0.406i)16-s + (0.447 − 4.26i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 − 0.703i)2-s + (0.967 + 0.253i)3-s + (−0.489 + 0.103i)4-s + (−0.306 − 0.687i)5-s + (0.106 − 0.699i)6-s + (0.585 + 0.810i)7-s + (0.109 + 0.336i)8-s + (0.871 + 0.490i)9-s + (−0.461 + 0.266i)10-s + (0.853 + 0.521i)11-s + (−0.499 − 0.0233i)12-s + (0.495 + 0.682i)13-s + (0.526 − 0.471i)14-s + (−0.121 − 0.743i)15-s + (0.228 − 0.101i)16-s + (0.108 − 1.03i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 + 0.539i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.842 + 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.80619 - 0.528513i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.80619 - 0.528513i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.67 - 0.438i)T \) |
| 7 | \( 1 + (-1.54 - 2.14i)T \) |
| 11 | \( 1 + (-2.83 - 1.72i)T \) |
| good | 5 | \( 1 + (0.684 + 1.53i)T + (-3.34 + 3.71i)T^{2} \) |
| 13 | \( 1 + (-1.78 - 2.46i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.447 + 4.26i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (0.153 - 0.720i)T + (-17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (6.96 + 4.01i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.76 + 8.50i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (5.99 + 2.66i)T + (20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (-4.23 - 4.70i)T + (-3.86 + 36.7i)T^{2} \) |
| 41 | \( 1 + (-0.907 - 2.79i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 6.89iT - 43T^{2} \) |
| 47 | \( 1 + (2.33 - 10.9i)T + (-42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (0.415 - 0.932i)T + (-35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (1.40 + 6.60i)T + (-53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (2.09 + 4.70i)T + (-40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (2.70 + 4.69i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (9.86 - 13.5i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (3.10 + 14.5i)T + (-66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (-1.58 + 0.166i)T + (77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (-0.791 - 0.575i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-3.79 - 2.19i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (11.6 - 8.50i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.10001406786558961730876638102, −9.689945377590412510668957943958, −9.347884361277990110417785241789, −8.385466903991720033619670249989, −7.83317537157284705185498270689, −6.25582778525118516554002324791, −4.62293086819465122046210023330, −4.22461622295898338427176977829, −2.68993879153957861046313216032, −1.58554170702075303537144518777,
1.47330711844501194627371593768, 3.46143656668680350192756434364, 3.99564350526585905950959912288, 5.66054256604076445130622759946, 6.84343300294713370685618545825, 7.44594334120880009888083435892, 8.325088842136045538012468973683, 8.975859753611495164392074622826, 10.25820822892755580767158029724, 10.86040933679157368672483535800
