| L(s) = 1 | + (−0.142 − 0.989i)2-s + (0.804 + 1.53i)3-s + (−0.959 + 0.281i)4-s + (1.14 + 1.92i)5-s + (1.40 − 1.01i)6-s + (−0.955 + 0.614i)7-s + (0.415 + 0.909i)8-s + (−1.70 + 2.46i)9-s + (1.73 − 1.40i)10-s + (−0.130 + 0.904i)11-s + (−1.20 − 1.24i)12-s + (2.47 − 3.84i)13-s + (0.744 + 0.858i)14-s + (−2.02 + 3.30i)15-s + (0.841 − 0.540i)16-s + (−1.74 + 5.94i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.699i)2-s + (0.464 + 0.885i)3-s + (−0.479 + 0.140i)4-s + (0.511 + 0.859i)5-s + (0.573 − 0.414i)6-s + (−0.361 + 0.232i)7-s + (0.146 + 0.321i)8-s + (−0.568 + 0.822i)9-s + (0.549 − 0.444i)10-s + (−0.0392 + 0.272i)11-s + (−0.347 − 0.359i)12-s + (0.685 − 1.06i)13-s + (0.198 + 0.229i)14-s + (−0.522 + 0.852i)15-s + (0.210 − 0.135i)16-s + (−0.423 + 1.44i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.178 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.178 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.13748 + 0.949305i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13748 + 0.949305i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 + 0.989i)T \) |
| 3 | \( 1 + (-0.804 - 1.53i)T \) |
| 5 | \( 1 + (-1.14 - 1.92i)T \) |
| 23 | \( 1 + (-1.02 + 4.68i)T \) |
| good | 7 | \( 1 + (0.955 - 0.614i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.130 - 0.904i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-2.47 + 3.84i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (1.74 - 5.94i)T + (-14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-0.111 - 0.378i)T + (-15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (2.85 - 9.72i)T + (-24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-3.71 - 8.14i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (4.47 + 5.16i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (7.60 + 6.59i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-1.93 + 4.23i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 8.91T + 47T^{2} \) |
| 53 | \( 1 + (-0.827 - 1.28i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (2.84 - 4.43i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-6.70 + 3.06i)T + (39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.882 - 6.13i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-9.58 + 1.37i)T + (68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-1.47 - 5.01i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-1.62 + 2.52i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-7.96 + 6.90i)T + (11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-6.57 + 14.4i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-10.6 + 12.3i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.50269134517389102901992696956, −10.20506706188828431586805836411, −8.895412605103091780134552479840, −8.591269120866988590458246200927, −7.22768010892587413309623540553, −6.02528509247417049316210127869, −5.11629363938929448797185463283, −3.73711095637741384673130265677, −3.12556020846495268581738184750, −1.98832791384959895004314847877,
0.77486725193173389209681421141, 2.19971459793575631343270599916, 3.76828035764873969439511816862, 4.96824370477604903646234778642, 6.10112131367179029946674762613, 6.67424682090617755842496548137, 7.71303598331335672593578721119, 8.441885937091331566657075172538, 9.413188951954182850197883006739, 9.627565557712490925574578003899
