L(s) = 1 | + (−0.707 + 0.707i)2-s + i·3-s − 1.00i·4-s + (−1.80 − 1.80i)5-s + (−0.707 − 0.707i)6-s + (1.93 + 1.80i)7-s + (0.707 + 0.707i)8-s − 9-s + 2.54·10-s + (0.936 + 0.936i)11-s + 1.00·12-s + (2 − 3i)13-s + (−2.64 + 0.0951i)14-s + (1.80 − 1.80i)15-s − 1.00·16-s − 0.496·17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + 0.577i·3-s − 0.500i·4-s + (−0.806 − 0.806i)5-s + (−0.288 − 0.288i)6-s + (0.732 + 0.681i)7-s + (0.250 + 0.250i)8-s − 0.333·9-s + 0.806·10-s + (0.282 + 0.282i)11-s + 0.288·12-s + (0.554 − 0.832i)13-s + (−0.706 + 0.0254i)14-s + (0.465 − 0.465i)15-s − 0.250·16-s − 0.120·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.255 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.255 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.832948 + 0.641653i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.832948 + 0.641653i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 - iT \) |
| 7 | \( 1 + (-1.93 - 1.80i)T \) |
| 13 | \( 1 + (-2 + 3i)T \) |
good | 5 | \( 1 + (1.80 + 1.80i)T + 5iT^{2} \) |
| 11 | \( 1 + (-0.936 - 0.936i)T + 11iT^{2} \) |
| 17 | \( 1 + 0.496T + 17T^{2} \) |
| 19 | \( 1 + (-3.07 - 3.07i)T + 19iT^{2} \) |
| 23 | \( 1 - 7.37iT - 23T^{2} \) |
| 29 | \( 1 - 7.25T + 29T^{2} \) |
| 31 | \( 1 + (-5.73 - 5.73i)T + 31iT^{2} \) |
| 37 | \( 1 + (-3.54 - 3.54i)T + 37iT^{2} \) |
| 41 | \( 1 + (6.11 + 6.11i)T + 41iT^{2} \) |
| 43 | \( 1 - 6.85iT - 43T^{2} \) |
| 47 | \( 1 + (5.87 - 5.87i)T - 47iT^{2} \) |
| 53 | \( 1 + 4.26T + 53T^{2} \) |
| 59 | \( 1 + (1.37 - 1.37i)T - 59iT^{2} \) |
| 61 | \( 1 + 0.496iT - 61T^{2} \) |
| 67 | \( 1 + (-9.11 + 9.11i)T - 67iT^{2} \) |
| 71 | \( 1 + (-11.3 + 11.3i)T - 71iT^{2} \) |
| 73 | \( 1 + (-3.54 + 3.54i)T - 73iT^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 + (-7.87 - 7.87i)T + 83iT^{2} \) |
| 89 | \( 1 + (-10.2 + 10.2i)T - 89iT^{2} \) |
| 97 | \( 1 + (-9.46 - 9.46i)T + 97iT^{2} \) |
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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
−10.93736374841672399771996861414, −9.911416643112221787890552552030, −9.103093261088814791395734381062, −8.115220222378413160863330821779, −7.973082453519033217210877215193, −6.38074221697596937558914220950, −5.27297793784732576491917971112, −4.64924460691354417162145725755, −3.29988634223920636584905149557, −1.28549466134834050305842340829,
0.885573481062164791498229567993, 2.45310539946190554623015387883, 3.67723885180427102129972312479, 4.66948122208777595483128561388, 6.52487236224739061619065303010, 7.04169583167147534468591811426, 8.070572145148027145131320694316, 8.600272228819305557660089345502, 9.904065715947696204390480197413, 10.84626958278381818994446329919