L(s) = 1 | + (−0.809 + 0.587i)2-s + (−0.347 − 1.06i)3-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)5-s + (0.909 + 0.660i)6-s + (−0.309 + 0.951i)7-s + (0.309 + 0.951i)8-s + (1.40 − 1.02i)9-s − 10-s + (−2.21 + 2.46i)11-s − 1.12·12-s + (−5.71 + 4.15i)13-s + (−0.309 − 0.951i)14-s + (0.347 − 1.06i)15-s + (−0.809 − 0.587i)16-s + (−2.78 − 2.02i)17-s + ⋯ |
L(s) = 1 | + (−0.572 + 0.415i)2-s + (−0.200 − 0.617i)3-s + (0.154 − 0.475i)4-s + (0.361 + 0.262i)5-s + (0.371 + 0.269i)6-s + (−0.116 + 0.359i)7-s + (0.109 + 0.336i)8-s + (0.467 − 0.340i)9-s − 0.316·10-s + (−0.668 + 0.743i)11-s − 0.324·12-s + (−1.58 + 1.15i)13-s + (−0.0825 − 0.254i)14-s + (0.0897 − 0.276i)15-s + (−0.202 − 0.146i)16-s + (−0.675 − 0.491i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.480 - 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.480 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.325076 + 0.548588i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.325076 + 0.548588i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 - 0.587i)T \) |
| 5 | \( 1 + (-0.809 - 0.587i)T \) |
| 7 | \( 1 + (0.309 - 0.951i)T \) |
| 11 | \( 1 + (2.21 - 2.46i)T \) |
good | 3 | \( 1 + (0.347 + 1.06i)T + (-2.42 + 1.76i)T^{2} \) |
| 13 | \( 1 + (5.71 - 4.15i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (2.78 + 2.02i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.06 - 3.26i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 3.63T + 23T^{2} \) |
| 29 | \( 1 + (2.68 - 8.26i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-3.65 + 2.65i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (0.241 - 0.742i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (1.55 + 4.77i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 5.39T + 43T^{2} \) |
| 47 | \( 1 + (-3.03 - 9.34i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (9.19 - 6.68i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (3.25 - 10.0i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (0.369 + 0.268i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 + (-5.51 - 4.00i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (4.67 - 14.3i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-13.1 + 9.54i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (5.98 + 4.35i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 4.97T + 89T^{2} \) |
| 97 | \( 1 + (2.33 - 1.69i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.35652668515318927233677983570, −9.588832982802863264337692317364, −9.110307762270660091481673946790, −7.74369429258738278942549497117, −7.08311202310667820117951061478, −6.61145825188995814272027724036, −5.40804726897598688016626986114, −4.50689444811586550692182681881, −2.63397794976757440853323025854, −1.64862663543279525337608781503,
0.38203433488891065566875038475, 2.22523913709542776026533992556, 3.31968441942573625006152313052, 4.69442963351669332111801377631, 5.28623008144686035996967733306, 6.66038521573094407748529967438, 7.68448533322973801259442193293, 8.356173476249345976165217482205, 9.585698198429319527169132525773, 9.951172425509350728555345559650