L(s) = 1 | + i·2-s + (0.707 + 0.707i)3-s − 4-s + (1 + i)5-s + (−0.707 + 0.707i)6-s + i·7-s − i·8-s + 1.00i·9-s + (−1 + i)10-s + (0.707 − 0.707i)11-s + (−0.707 − 0.707i)12-s + (−0.707 − 0.707i)13-s − 14-s + 1.41i·15-s + 16-s − 17-s + ⋯ |
L(s) = 1 | + i·2-s + (0.707 + 0.707i)3-s − 4-s + (1 + i)5-s + (−0.707 + 0.707i)6-s + i·7-s − i·8-s + 1.00i·9-s + (−1 + i)10-s + (0.707 − 0.707i)11-s + (−0.707 − 0.707i)12-s + (−0.707 − 0.707i)13-s − 14-s + 1.41i·15-s + 16-s − 17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.509207006\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.509207006\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 37 | \( 1 + (0.707 - 0.707i)T \) |
good | 5 | \( 1 + (-1 - i)T + iT^{2} \) |
| 7 | \( 1 - iT - T^{2} \) |
| 11 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 13 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 23 | \( 1 + iT - T^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 + 1.41iT - T^{2} \) |
| 41 | \( 1 - 1.41T + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + 1.41iT - T^{2} \) |
| 53 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 59 | \( 1 + (1 + i)T + iT^{2} \) |
| 61 | \( 1 + iT^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + 1.41T + T^{2} \) |
| 73 | \( 1 - iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 89 | \( 1 - iT - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.698890298072954460877870737396, −8.958435820172427331157206344924, −8.393014511717577178454388738189, −7.47632081438894561562739199847, −6.51916864735104308332892407618, −5.84993479950917938516597283666, −5.20125335017672621399035988200, −4.07212914319151142534870031858, −3.03767223399095601635880928603, −2.23221680913406773634352888646,
1.20350202001712558603317857059, 1.80472073409812947468331512827, 2.87290515009225773391670960859, 4.15046870717702239151705133913, 4.67338162506538458044177612294, 5.80598979799368224361968317336, 7.00952058170054677866723705376, 7.52615655174249543087628468813, 8.856479840603154809450966740309, 9.176783102513466926566882858029