| L(s) = 1 | + (−0.772 − 0.0311i)2-s + (−0.278 + 0.960i)3-s + (−1.39 − 0.112i)4-s + (3.22 + 1.22i)5-s + (0.244 − 0.733i)6-s + (−1.25 − 2.93i)7-s + (2.61 + 0.317i)8-s + (−0.845 − 0.534i)9-s + (−2.44 − 1.04i)10-s + (−1.20 − 1.90i)11-s + (0.497 − 1.31i)12-s + (−0.485 − 3.57i)13-s + (0.874 + 2.30i)14-s + (−2.06 + 2.75i)15-s + (0.761 + 0.123i)16-s + (4.45 − 1.89i)17-s + ⋯ |
| L(s) = 1 | + (−0.546 − 0.0220i)2-s + (−0.160 + 0.554i)3-s + (−0.698 − 0.0564i)4-s + (1.44 + 0.546i)5-s + (0.0999 − 0.299i)6-s + (−0.472 − 1.10i)7-s + (0.923 + 0.112i)8-s + (−0.281 − 0.178i)9-s + (−0.774 − 0.330i)10-s + (−0.363 − 0.574i)11-s + (0.143 − 0.378i)12-s + (−0.134 − 0.990i)13-s + (0.233 + 0.616i)14-s + (−0.534 + 0.711i)15-s + (0.190 + 0.0309i)16-s + (1.07 − 0.459i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 + 0.461i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.887 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.968774 - 0.236968i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.968774 - 0.236968i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.278 - 0.960i)T \) |
| 13 | \( 1 + (0.485 + 3.57i)T \) |
| good | 2 | \( 1 + (0.772 + 0.0311i)T + (1.99 + 0.160i)T^{2} \) |
| 5 | \( 1 + (-3.22 - 1.22i)T + (3.74 + 3.31i)T^{2} \) |
| 7 | \( 1 + (1.25 + 2.93i)T + (-4.84 + 5.04i)T^{2} \) |
| 11 | \( 1 + (1.20 + 1.90i)T + (-4.71 + 9.93i)T^{2} \) |
| 17 | \( 1 + (-4.45 + 1.89i)T + (11.7 - 12.2i)T^{2} \) |
| 19 | \( 1 + (0.107 - 0.0619i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.94 - 3.37i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.346 + 8.60i)T + (-28.9 - 2.33i)T^{2} \) |
| 31 | \( 1 + (-2.80 - 3.16i)T + (-3.73 + 30.7i)T^{2} \) |
| 37 | \( 1 + (-6.65 + 1.35i)T + (34.0 - 14.5i)T^{2} \) |
| 41 | \( 1 + (-8.89 - 2.57i)T + (34.6 + 21.9i)T^{2} \) |
| 43 | \( 1 + (0.0652 - 0.319i)T + (-39.5 - 16.8i)T^{2} \) |
| 47 | \( 1 + (-3.41 + 2.35i)T + (16.6 - 43.9i)T^{2} \) |
| 53 | \( 1 + (-1.17 + 9.64i)T + (-51.4 - 12.6i)T^{2} \) |
| 59 | \( 1 + (-0.526 - 3.24i)T + (-55.9 + 18.6i)T^{2} \) |
| 61 | \( 1 + (-2.23 + 1.67i)T + (16.9 - 58.5i)T^{2} \) |
| 67 | \( 1 + (0.395 + 4.89i)T + (-66.1 + 10.7i)T^{2} \) |
| 71 | \( 1 + (7.65 - 7.34i)T + (2.85 - 70.9i)T^{2} \) |
| 73 | \( 1 + (7.73 - 14.7i)T + (-41.4 - 60.0i)T^{2} \) |
| 79 | \( 1 + (-4.19 - 6.07i)T + (-28.0 + 73.8i)T^{2} \) |
| 83 | \( 1 + (-1.34 + 5.46i)T + (-73.4 - 38.5i)T^{2} \) |
| 89 | \( 1 + (11.0 + 6.39i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.35 + 5.19i)T + (19.4 + 95.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.32913693291504671354127343969, −9.976449076617488588597730688936, −9.547615607902987358904385270428, −8.235066221213376827605284798974, −7.35866720656752421399543461597, −5.99541359081901504686895937255, −5.38498339161038327148316873611, −4.04181055993413186824197408913, −2.83717801197500141029793482598, −0.840294003528414385998526742352,
1.41415029906009987164111240475, 2.53140871706245946942244855011, 4.52792784682327794704068346267, 5.55696963017999096361763917975, 6.19484826400261539442585389171, 7.48139569667698773756100426832, 8.582945036000227877642221694541, 9.250017149421547296750636964432, 9.766473308788158979402550576123, 10.67046237034735135591962934189
