L(s) = 1 | + (0.707 + 0.707i)2-s + 1.00i·4-s + (−0.707 + 0.707i)5-s + (−0.707 + 0.707i)8-s − 1.00·10-s − 1.00·16-s + 1.41i·17-s + (1 + i)19-s + (−0.707 − 0.707i)20-s − 1.41i·23-s − 1.00i·25-s − 2i·31-s + (−0.707 − 0.707i)32-s + (−1.00 + 1.00i)34-s + 1.41i·38-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s + 1.00i·4-s + (−0.707 + 0.707i)5-s + (−0.707 + 0.707i)8-s − 1.00·10-s − 1.00·16-s + 1.41i·17-s + (1 + i)19-s + (−0.707 − 0.707i)20-s − 1.41i·23-s − 1.00i·25-s − 2i·31-s + (−0.707 − 0.707i)32-s + (−1.00 + 1.00i)34-s + 1.41i·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.172592230\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.172592230\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
good | 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - 1.41iT - T^{2} \) |
| 19 | \( 1 + (-1 - i)T + iT^{2} \) |
| 23 | \( 1 + 1.41iT - T^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 + 2iT - T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 - 1.41T + T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + (1 - i)T - iT^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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
−10.97317434809376828717867320608, −10.18655162890125860536696265526, −8.873034328356344227895818577097, −7.970036923532743868076785153349, −7.45237123227378816983632751686, −6.36693237660249894171426637406, −5.74710953651620593703347171559, −4.32768987079081907197032935784, −3.71103122058905116286250460304, −2.51040726453255872723452292787,
1.13445124268816426143868428178, 2.83457145538099995172097371699, 3.76971133479913460119449386306, 4.95770460266557193651478883978, 5.34522420386373569232417709751, 6.85744546489755928801315468096, 7.62033086590413236406189592231, 9.042360657172713694227520769634, 9.383612710503227533419508223958, 10.58622698184824029020416472264