| L(s) = 1 | + (−1.39 + 0.210i)2-s + (1.49 − 0.878i)3-s + (1.91 − 0.587i)4-s + (1.84 + 0.844i)5-s + (−1.90 + 1.54i)6-s + (−2.22 + 2.32i)7-s + (−2.54 + 1.22i)8-s + (1.45 − 2.62i)9-s + (−2.76 − 0.792i)10-s + (−5.66 + 0.540i)11-s + (2.33 − 2.55i)12-s + (1.43 − 0.276i)13-s + (2.61 − 3.72i)14-s + (3.50 − 0.365i)15-s + (3.30 − 2.24i)16-s + (2.25 + 4.38i)17-s + ⋯ |
| L(s) = 1 | + (−0.988 + 0.148i)2-s + (0.861 − 0.507i)3-s + (0.955 − 0.293i)4-s + (0.827 + 0.377i)5-s + (−0.776 + 0.629i)6-s + (−0.839 + 0.880i)7-s + (−0.901 + 0.432i)8-s + (0.484 − 0.874i)9-s + (−0.874 − 0.250i)10-s + (−1.70 + 0.163i)11-s + (0.674 − 0.738i)12-s + (0.397 − 0.0766i)13-s + (0.699 − 0.995i)14-s + (0.904 − 0.0942i)15-s + (0.827 − 0.561i)16-s + (0.547 + 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.25417 + 0.507327i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25417 + 0.507327i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 - 0.210i)T \) |
| 3 | \( 1 + (-1.49 + 0.878i)T \) |
| 67 | \( 1 + (8.01 + 1.65i)T \) |
| good | 5 | \( 1 + (-1.84 - 0.844i)T + (3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (2.22 - 2.32i)T + (-0.333 - 6.99i)T^{2} \) |
| 11 | \( 1 + (5.66 - 0.540i)T + (10.8 - 2.08i)T^{2} \) |
| 13 | \( 1 + (-1.43 + 0.276i)T + (12.0 - 4.83i)T^{2} \) |
| 17 | \( 1 + (-2.25 - 4.38i)T + (-9.86 + 13.8i)T^{2} \) |
| 19 | \( 1 + (-5.85 - 6.14i)T + (-0.904 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-7.14 - 2.86i)T + (16.6 + 15.8i)T^{2} \) |
| 29 | \( 1 + (-0.448 + 0.258i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.08 - 5.63i)T + (-28.7 - 11.5i)T^{2} \) |
| 37 | \( 1 + (-3.45 + 5.97i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-9.17 - 0.437i)T + (40.8 + 3.89i)T^{2} \) |
| 43 | \( 1 + (0.737 - 1.14i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (-5.37 + 4.22i)T + (11.0 - 45.6i)T^{2} \) |
| 53 | \( 1 + (-0.952 - 1.48i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (2.95 - 3.41i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (5.41 + 0.516i)T + (59.8 + 11.5i)T^{2} \) |
| 71 | \( 1 + (4.77 + 2.46i)T + (41.1 + 57.8i)T^{2} \) |
| 73 | \( 1 + (11.5 + 1.10i)T + (71.6 + 13.8i)T^{2} \) |
| 79 | \( 1 + (8.67 + 3.00i)T + (62.0 + 48.8i)T^{2} \) |
| 83 | \( 1 + (-3.55 - 4.98i)T + (-27.1 + 78.4i)T^{2} \) |
| 89 | \( 1 + (3.47 + 0.499i)T + (85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (3.99 - 6.91i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.11783436476700108062284663449, −9.460191158128642226719616074064, −8.741190673018255486064196390108, −7.80539952246957089604554230191, −7.25289339206003601957914974586, −5.91473767624388743509312457039, −5.73344432951542624573930687965, −3.22273501520124083307827836830, −2.66149236802869221613419116501, −1.50698958051327042760714594834,
0.877739136303939548804685458120, 2.68700268841447042776304712658, 3.08121050845480936193671718365, 4.74571760153767887457463556059, 5.77845223552098858932197946248, 7.25121203754039411006647424912, 7.53006226487953148604219715270, 8.750043804701760853420891491661, 9.450989650576636422972917728411, 9.879212152783358264444001154413
