L(s) = 1 | + (0.421 + 1.34i)2-s + (0.707 − 0.707i)3-s + (−1.64 + 1.13i)4-s + (2.44 + 2.44i)5-s + (1.25 + 0.656i)6-s − i·7-s + (−2.22 − 1.73i)8-s − 1.00i·9-s + (−2.26 + 4.33i)10-s + (2.14 + 2.14i)11-s + (−0.357 + 1.96i)12-s + (−3.07 + 3.07i)13-s + (1.34 − 0.421i)14-s + 3.45·15-s + (1.40 − 3.74i)16-s + 1.91·17-s + ⋯ |
L(s) = 1 | + (0.298 + 0.954i)2-s + (0.408 − 0.408i)3-s + (−0.822 + 0.569i)4-s + (1.09 + 1.09i)5-s + (0.511 + 0.267i)6-s − 0.377i·7-s + (−0.788 − 0.615i)8-s − 0.333i·9-s + (−0.717 + 1.36i)10-s + (0.645 + 0.645i)11-s + (−0.103 + 0.568i)12-s + (−0.852 + 0.852i)13-s + (0.360 − 0.112i)14-s + 0.892·15-s + (0.352 − 0.935i)16-s + 0.463·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0328 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0328 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.28044 + 1.32321i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.28044 + 1.32321i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.421 - 1.34i)T \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + (-2.44 - 2.44i)T + 5iT^{2} \) |
| 11 | \( 1 + (-2.14 - 2.14i)T + 11iT^{2} \) |
| 13 | \( 1 + (3.07 - 3.07i)T - 13iT^{2} \) |
| 17 | \( 1 - 1.91T + 17T^{2} \) |
| 19 | \( 1 + (-1.94 + 1.94i)T - 19iT^{2} \) |
| 23 | \( 1 + 5.41iT - 23T^{2} \) |
| 29 | \( 1 + (4.71 - 4.71i)T - 29iT^{2} \) |
| 31 | \( 1 + 4.00T + 31T^{2} \) |
| 37 | \( 1 + (-0.294 - 0.294i)T + 37iT^{2} \) |
| 41 | \( 1 + 10.1iT - 41T^{2} \) |
| 43 | \( 1 + (-3.10 - 3.10i)T + 43iT^{2} \) |
| 47 | \( 1 - 7.75T + 47T^{2} \) |
| 53 | \( 1 + (3.04 + 3.04i)T + 53iT^{2} \) |
| 59 | \( 1 + (9.93 + 9.93i)T + 59iT^{2} \) |
| 61 | \( 1 + (-6.60 + 6.60i)T - 61iT^{2} \) |
| 67 | \( 1 + (-7.34 + 7.34i)T - 67iT^{2} \) |
| 71 | \( 1 + 6.77iT - 71T^{2} \) |
| 73 | \( 1 - 2.35iT - 73T^{2} \) |
| 79 | \( 1 + 9.34T + 79T^{2} \) |
| 83 | \( 1 + (-11.5 + 11.5i)T - 83iT^{2} \) |
| 89 | \( 1 + 3.57iT - 89T^{2} \) |
| 97 | \( 1 + 5.98T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.10079121292566182217734594706, −10.70885053663430227687327801174, −9.591275966250547796725762194699, −9.115379450896087765288660631618, −7.54934654883547230036124764559, −6.96004429582413169120439506152, −6.29149527338444651063547583772, −5.00217869995138151124187190637, −3.60239810041980465160136321900, −2.20046656517100582882972049846,
1.37586803146084657513373169904, 2.75442141803661846539260886189, 4.05948047778441277994255009726, 5.41385545136135178816958199997, 5.71378965865158224698960850637, 7.902720341845153411092104213708, 9.008833598503471013683033693472, 9.513672026991046301631761508419, 10.17640057356635827577290663461, 11.40551691885353570887537038144