L(s) = 1 | + (−1 − i)2-s + (1.87 − 1.87i)3-s + 2i·4-s − 3i·5-s − 3.74·6-s + 3.74i·7-s + (2 − 2i)8-s − 4i·9-s + (−3 + 3i)10-s + (−1.87 + 1.87i)11-s + (3.74 + 3.74i)12-s + i·13-s + (3.74 − 3.74i)14-s + (−5.61 − 5.61i)15-s − 4·16-s + (−2 − 2i)17-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + (1.08 − 1.08i)3-s + i·4-s − 1.34i·5-s − 1.52·6-s + 1.41i·7-s + (0.707 − 0.707i)8-s − 1.33i·9-s + (−0.948 + 0.948i)10-s + (−0.564 + 0.564i)11-s + (1.08 + 1.08i)12-s + 0.277i·13-s + (0.999 − 0.999i)14-s + (−1.44 − 1.44i)15-s − 16-s + (−0.485 − 0.485i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.638010 - 0.772603i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.638010 - 0.772603i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + i)T \) |
| 29 | \( 1 + (-5 - 2i)T \) |
good | 3 | \( 1 + (-1.87 + 1.87i)T - 3iT^{2} \) |
| 5 | \( 1 + 3iT - 5T^{2} \) |
| 7 | \( 1 - 3.74iT - 7T^{2} \) |
| 11 | \( 1 + (1.87 - 1.87i)T - 11iT^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 + (2 + 2i)T + 17iT^{2} \) |
| 19 | \( 1 + (-3.74 + 3.74i)T - 19iT^{2} \) |
| 23 | \( 1 - 7.48iT - 23T^{2} \) |
| 31 | \( 1 + (1.87 - 1.87i)T - 31iT^{2} \) |
| 37 | \( 1 - 37iT^{2} \) |
| 41 | \( 1 + (-1 + i)T - 41iT^{2} \) |
| 43 | \( 1 + (-1.87 + 1.87i)T - 43iT^{2} \) |
| 47 | \( 1 + (5.61 + 5.61i)T + 47iT^{2} \) |
| 53 | \( 1 + 11T + 53T^{2} \) |
| 59 | \( 1 + 7.48iT - 59T^{2} \) |
| 61 | \( 1 + (-6 - 6i)T + 61iT^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 + 3.74T + 71T^{2} \) |
| 73 | \( 1 + (-4 + 4i)T - 73iT^{2} \) |
| 79 | \( 1 + (5.61 - 5.61i)T - 79iT^{2} \) |
| 83 | \( 1 - 7.48iT - 83T^{2} \) |
| 89 | \( 1 + (3 + 3i)T + 89iT^{2} \) |
| 97 | \( 1 + (5 - 5i)T - 97iT^{2} \) |
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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
−12.99807736556550127613517443140, −12.35631793917628635408436672923, −11.57556985252288971744742865287, −9.500668795887163569720153386861, −8.968118188040218047164649629676, −8.204998988594760120655580243131, −7.15789808710365345146737248894, −5.04650914494567174551151981249, −2.87200882171538248392563210833, −1.66036019463517297913566076762,
2.99193727525941072984503254440, 4.38461096992971305225468870703, 6.32881902144624448023266432763, 7.55350939129534095624651835145, 8.351683652848722981564315683558, 9.769016217403337002211538135586, 10.46539479919560384146049423005, 10.90295738501258253132982673149, 13.50666180729718310602795263306, 14.35176403047783509123493032045