| L(s) = 1 | + (1.09 − 0.895i)2-s + (1.12 + 0.926i)3-s + (0.395 − 1.96i)4-s + (−1.23 − 4.05i)5-s + (2.06 + 0.00292i)6-s + (−0.980 − 0.195i)7-s + (−1.32 − 2.50i)8-s + (−0.169 − 0.851i)9-s + (−4.98 − 3.33i)10-s + (−0.421 + 4.28i)11-s + (2.26 − 1.84i)12-s + (0.651 + 0.197i)13-s + (−1.24 + 0.664i)14-s + (2.36 − 5.72i)15-s + (−3.68 − 1.55i)16-s + (1.40 + 3.38i)17-s + ⋯ |
| L(s) = 1 | + (0.773 − 0.633i)2-s + (0.651 + 0.534i)3-s + (0.197 − 0.980i)4-s + (−0.550 − 1.81i)5-s + (0.842 + 0.00119i)6-s + (−0.370 − 0.0737i)7-s + (−0.467 − 0.883i)8-s + (−0.0564 − 0.283i)9-s + (−1.57 − 1.05i)10-s + (−0.127 + 1.29i)11-s + (0.653 − 0.532i)12-s + (0.180 + 0.0547i)13-s + (−0.333 + 0.177i)14-s + (0.611 − 1.47i)15-s + (−0.921 − 0.387i)16-s + (0.340 + 0.821i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.747 + 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.816207 - 2.14763i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.816207 - 2.14763i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.09 + 0.895i)T \) |
| 7 | \( 1 + (0.980 + 0.195i)T \) |
| good | 3 | \( 1 + (-1.12 - 0.926i)T + (0.585 + 2.94i)T^{2} \) |
| 5 | \( 1 + (1.23 + 4.05i)T + (-4.15 + 2.77i)T^{2} \) |
| 11 | \( 1 + (0.421 - 4.28i)T + (-10.7 - 2.14i)T^{2} \) |
| 13 | \( 1 + (-0.651 - 0.197i)T + (10.8 + 7.22i)T^{2} \) |
| 17 | \( 1 + (-1.40 - 3.38i)T + (-12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (2.02 + 3.78i)T + (-10.5 + 15.7i)T^{2} \) |
| 23 | \( 1 + (-4.91 + 7.35i)T + (-8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (3.85 - 0.380i)T + (28.4 - 5.65i)T^{2} \) |
| 31 | \( 1 + (1.16 + 1.16i)T + 31iT^{2} \) |
| 37 | \( 1 + (-7.61 - 4.07i)T + (20.5 + 30.7i)T^{2} \) |
| 41 | \( 1 + (3.24 + 2.16i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-0.615 + 0.505i)T + (8.38 - 42.1i)T^{2} \) |
| 47 | \( 1 + (-7.24 + 3.00i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-5.87 - 0.578i)T + (51.9 + 10.3i)T^{2} \) |
| 59 | \( 1 + (-2.04 + 0.620i)T + (49.0 - 32.7i)T^{2} \) |
| 61 | \( 1 + (4.40 - 5.36i)T + (-11.9 - 59.8i)T^{2} \) |
| 67 | \( 1 + (-0.487 + 0.594i)T + (-13.0 - 65.7i)T^{2} \) |
| 71 | \( 1 + (0.169 - 0.853i)T + (-65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-10.5 + 2.10i)T + (67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (1.37 + 0.571i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-10.5 + 5.64i)T + (46.1 - 69.0i)T^{2} \) |
| 89 | \( 1 + (4.38 + 6.56i)T + (-34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (4.64 + 4.64i)T + 97iT^{2} \) |
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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
−9.721577136422054959758967847360, −9.049108244443463637878887757007, −8.521229221489327319192733534794, −7.22962759831232290675438735879, −6.04342273232502745636318774335, −4.84029569495785515706816673141, −4.40420127138795763114789665131, −3.61883381911842424045895074346, −2.25617904269686852203513973019, −0.78355048444043395079027544245,
2.44808203358492297944530043009, 3.21071607279998566536152134583, 3.73406571027377160327223907622, 5.46816696850667656048405997630, 6.22796236484942124589466837838, 7.16336036505290879729497196003, 7.61514847121554309952334922886, 8.312363181624347591736849491920, 9.442147578381326766166728666756, 10.86890963820125418114907742834
