L(s) = 1 | + 2-s + (0.707 − 0.707i)3-s + 4-s + (−2.22 − 0.248i)5-s + (0.707 − 0.707i)6-s + (−2.88 + 2.88i)7-s + 8-s − 1.00i·9-s + (−2.22 − 0.248i)10-s − 4.93i·11-s + (0.707 − 0.707i)12-s − 1.83·13-s + (−2.88 + 2.88i)14-s + (−1.74 + 1.39i)15-s + 16-s − 2.74i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.408 − 0.408i)3-s + 0.5·4-s + (−0.993 − 0.111i)5-s + (0.288 − 0.288i)6-s + (−1.09 + 1.09i)7-s + 0.353·8-s − 0.333i·9-s + (−0.702 − 0.0787i)10-s − 1.48i·11-s + (0.204 − 0.204i)12-s − 0.507·13-s + (−0.772 + 0.772i)14-s + (−0.451 + 0.360i)15-s + 0.250·16-s − 0.665i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.875 + 0.482i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.875 + 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8810926190\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8810926190\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (2.22 + 0.248i)T \) |
| 37 | \( 1 + (-5.38 - 2.83i)T \) |
good | 7 | \( 1 + (2.88 - 2.88i)T - 7iT^{2} \) |
| 11 | \( 1 + 4.93iT - 11T^{2} \) |
| 13 | \( 1 + 1.83T + 13T^{2} \) |
| 17 | \( 1 + 2.74iT - 17T^{2} \) |
| 19 | \( 1 + (5.44 + 5.44i)T + 19iT^{2} \) |
| 23 | \( 1 + 8.34T + 23T^{2} \) |
| 29 | \( 1 + (-0.469 + 0.469i)T - 29iT^{2} \) |
| 31 | \( 1 + (-2.48 - 2.48i)T + 31iT^{2} \) |
| 41 | \( 1 + 4.25iT - 41T^{2} \) |
| 43 | \( 1 + 5.61T + 43T^{2} \) |
| 47 | \( 1 + (0.544 - 0.544i)T - 47iT^{2} \) |
| 53 | \( 1 + (7.02 + 7.02i)T + 53iT^{2} \) |
| 59 | \( 1 + (-7.44 - 7.44i)T + 59iT^{2} \) |
| 61 | \( 1 + (-1.23 - 1.23i)T + 61iT^{2} \) |
| 67 | \( 1 + (5.74 + 5.74i)T + 67iT^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + (5.94 - 5.94i)T - 73iT^{2} \) |
| 79 | \( 1 + (4.75 + 4.75i)T + 79iT^{2} \) |
| 83 | \( 1 + (0.579 + 0.579i)T + 83iT^{2} \) |
| 89 | \( 1 + (-2.52 + 2.52i)T - 89iT^{2} \) |
| 97 | \( 1 - 17.4iT - 97T^{2} \) |
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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
−9.293544752485790975798128840320, −8.564563830088887243353774092686, −7.925628603897651345699225159406, −6.73318279530020908768863072505, −6.25492404269705541418284279905, −5.20189057823703669337367427527, −4.04657782954543394431168465420, −3.10187739532776001729175861331, −2.46725363945999767268653470207, −0.26663165534856990021972355105,
2.06081292017727190801873054767, 3.36977613736369156273229120141, 4.18740765242430048609728062306, 4.43496177966205006383067057556, 6.07641280048425006310526884587, 6.85328507721615771915400620066, 7.66875231545292215931267532785, 8.253542824728251846378975170778, 9.838999716138258674201765809405, 10.03044568593519318093626720279