L(s) = 1 | + (−1.29 + 0.747i)2-s + (−0.0473 − 0.0820i)3-s + (0.118 − 0.204i)4-s − i·5-s + (0.122 + 0.0708i)6-s + (4.18 + 2.41i)7-s − 2.63i·8-s + (1.49 − 2.59i)9-s + (0.747 + 1.29i)10-s + (0.926 − 0.534i)11-s − 0.0224·12-s − 7.21·14-s + (−0.0820 + 0.0473i)15-s + (2.20 + 3.82i)16-s + (1.77 − 3.08i)17-s + 4.47i·18-s + ⋯ |
L(s) = 1 | + (−0.915 + 0.528i)2-s + (−0.0273 − 0.0473i)3-s + (0.0591 − 0.102i)4-s − 0.447i·5-s + (0.0501 + 0.0289i)6-s + (1.57 + 0.912i)7-s − 0.932i·8-s + (0.498 − 0.863i)9-s + (0.236 + 0.409i)10-s + (0.279 − 0.161i)11-s − 0.00647·12-s − 1.92·14-s + (−0.0211 + 0.0122i)15-s + (0.552 + 0.956i)16-s + (0.431 − 0.747i)17-s + 1.05i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.03375 - 0.132726i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.03375 - 0.132726i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + iT \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (1.29 - 0.747i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (0.0473 + 0.0820i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (-4.18 - 2.41i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.926 + 0.534i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.77 + 3.08i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.96 + 2.86i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.54 + 6.13i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.736 + 1.27i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 1.46iT - 31T^{2} \) |
| 37 | \( 1 + (-0.0219 + 0.0126i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.232 + 0.133i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.77 + 3.08i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 6.51iT - 47T^{2} \) |
| 53 | \( 1 - 0.991T + 53T^{2} \) |
| 59 | \( 1 + (-7.55 - 4.36i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.16 - 5.48i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.48 + 2.58i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.72 - 3.88i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 10.1iT - 73T^{2} \) |
| 79 | \( 1 - 8.78T + 79T^{2} \) |
| 83 | \( 1 + 0.725iT - 83T^{2} \) |
| 89 | \( 1 + (11.6 - 6.75i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.97 - 1.71i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.862249937897033273011028385000, −8.967591662193905578058880869623, −8.586657947729784655667640038793, −7.85087719615188249286732441603, −6.90255542193932481844117410864, −5.97948742169194886376911192381, −4.81392116337144513752694829147, −3.97361344777388104785801931470, −2.19604894370553062122683139451, −0.78384325118826629933454676446,
1.42681899750208307142342359318, 2.02204097091666743621171475506, 3.88503630756856136380133426187, 4.77213501532738912871415041970, 5.76729223345913630138509718090, 7.19087638846557101956728945213, 8.000985988716756314295050319801, 8.321468436276314933046970775077, 9.684314787211653270192208015995, 10.28659706874373253273694740700