L(s) = 1 | + i·2-s + (1 − i)3-s − 4-s + (1 + i)6-s − i·8-s − i·9-s + (−1 − i)11-s + (−1 + i)12-s + 16-s + 18-s − 2i·19-s + (1 − i)22-s + (−1 − i)24-s − i·25-s + i·32-s − 2·33-s + ⋯ |
L(s) = 1 | + i·2-s + (1 − i)3-s − 4-s + (1 + i)6-s − i·8-s − i·9-s + (−1 − i)11-s + (−1 + i)12-s + 16-s + 18-s − 2i·19-s + (1 − i)22-s + (−1 − i)24-s − i·25-s + i·32-s − 2·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.321992698\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.321992698\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 17 | \( 1 \) |
good | 3 | \( 1 + (-1 + i)T - iT^{2} \) |
| 5 | \( 1 + iT^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (1 + i)T + iT^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 19 | \( 1 + 2iT - T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + iT^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (-1 - i)T + iT^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - iT^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + iT^{2} \) |
| 73 | \( 1 + (-1 + i)T - iT^{2} \) |
| 79 | \( 1 - iT^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (1 - i)T - iT^{2} \) |
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
−8.879878327375682117319806791480, −8.061037241949871218661066524175, −7.76950220363082772312762822276, −6.86612811204264592196064233154, −6.28203709917504211777574357983, −5.28861005223960289897663629154, −4.43584965500702648356072977155, −3.14631737396415544365951699766, −2.49843459899612704812075948417, −0.821569623257130770884288497517,
1.77402779539533271587330371493, 2.62730330798888613834246385901, 3.56474124342709356998305094179, 4.10674858456214283759283665622, 5.01251587117738563151859986688, 5.74100164481167738636768386791, 7.33456217188006719694955678008, 8.075760031026605745351206774004, 8.660174787931424491343480036435, 9.578716163200738834477334255141