| L(s) = 1 | + (0.841 + 0.540i)2-s + (−0.142 − 0.989i)3-s + (0.415 + 0.909i)4-s + (0.959 + 0.281i)5-s + (0.415 − 0.909i)6-s + (0.730 − 0.843i)7-s + (−0.142 + 0.989i)8-s + (−0.959 + 0.281i)9-s + (0.654 + 0.755i)10-s + (0.307 − 0.197i)11-s + (0.841 − 0.540i)12-s + (4.06 + 4.68i)13-s + (1.07 − 0.314i)14-s + (0.142 − 0.989i)15-s + (−0.654 + 0.755i)16-s + (1.64 − 3.59i)17-s + ⋯ |
| L(s) = 1 | + (0.594 + 0.382i)2-s + (−0.0821 − 0.571i)3-s + (0.207 + 0.454i)4-s + (0.429 + 0.125i)5-s + (0.169 − 0.371i)6-s + (0.276 − 0.318i)7-s + (−0.0503 + 0.349i)8-s + (−0.319 + 0.0939i)9-s + (0.207 + 0.238i)10-s + (0.0927 − 0.0596i)11-s + (0.242 − 0.156i)12-s + (1.12 + 1.30i)13-s + (0.286 − 0.0839i)14-s + (0.0367 − 0.255i)15-s + (−0.163 + 0.188i)16-s + (0.398 − 0.872i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.33751 + 0.258235i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.33751 + 0.258235i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.841 - 0.540i)T \) |
| 3 | \( 1 + (0.142 + 0.989i)T \) |
| 5 | \( 1 + (-0.959 - 0.281i)T \) |
| 23 | \( 1 + (-4.43 - 1.83i)T \) |
| good | 7 | \( 1 + (-0.730 + 0.843i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-0.307 + 0.197i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-4.06 - 4.68i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.64 + 3.59i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (0.805 + 1.76i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-2.03 + 4.46i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.0682 + 0.474i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (6.87 - 2.01i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-11.1 - 3.28i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (1.18 + 8.26i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 + (1.51 - 1.75i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (0.708 + 0.818i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (1.05 - 7.31i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (3.02 + 1.94i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (8.18 + 5.25i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (0.843 + 1.84i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-8.57 - 9.89i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-4.74 + 1.39i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (1.60 + 11.1i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (16.9 + 4.97i)T + (81.6 + 52.4i)T^{2} \) |
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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
−10.79251937965560943191847060772, −9.455886524377294584140357539191, −8.706386408446362035282841209512, −7.63457585701538751444166069127, −6.82760712823584288134414148525, −6.17580946381119444670884734475, −5.12635194246141995300718371087, −4.09835715247241886138446880132, −2.82628580361362527783526738646, −1.43773853555935490573050878455,
1.38367985764901053381195913837, 2.93183909595906995640760446956, 3.82647374496321100447540230125, 5.03545935127458233863238043225, 5.70130094125404948513398019596, 6.53981329518141988056028530208, 8.047412437250001821593167267272, 8.779558857177780538349419252445, 9.786406328938917945255988048263, 10.70788562193321600349798340225
