L(s) = 1 | + (−0.726 + 1.57i)3-s + (−2.20 − 0.341i)5-s + (−2.06 − 1.64i)7-s + (−1.94 − 2.28i)9-s − 1.06i·11-s + 4.82·13-s + (2.14 − 3.22i)15-s + 7.89i·17-s − 4.02i·19-s + (4.09 − 2.05i)21-s + 5.69·23-s + (4.76 + 1.50i)25-s + (5.00 − 1.40i)27-s + 2.00i·29-s − 4.89i·31-s + ⋯ |
L(s) = 1 | + (−0.419 + 0.907i)3-s + (−0.988 − 0.152i)5-s + (−0.781 − 0.623i)7-s + (−0.648 − 0.761i)9-s − 0.320i·11-s + 1.33·13-s + (0.552 − 0.833i)15-s + 1.91i·17-s − 0.922i·19-s + (0.893 − 0.448i)21-s + 1.18·23-s + (0.953 + 0.301i)25-s + (0.962 − 0.269i)27-s + 0.372i·29-s − 0.879i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.306i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 - 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.957758 + 0.150528i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.957758 + 0.150528i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.726 - 1.57i)T \) |
| 5 | \( 1 + (2.20 + 0.341i)T \) |
| 7 | \( 1 + (2.06 + 1.64i)T \) |
good | 11 | \( 1 + 1.06iT - 11T^{2} \) |
| 13 | \( 1 - 4.82T + 13T^{2} \) |
| 17 | \( 1 - 7.89iT - 17T^{2} \) |
| 19 | \( 1 + 4.02iT - 19T^{2} \) |
| 23 | \( 1 - 5.69T + 23T^{2} \) |
| 29 | \( 1 - 2.00iT - 29T^{2} \) |
| 31 | \( 1 + 4.89iT - 31T^{2} \) |
| 37 | \( 1 + 2.56iT - 37T^{2} \) |
| 41 | \( 1 - 5.08T + 41T^{2} \) |
| 43 | \( 1 - 6.15iT - 43T^{2} \) |
| 47 | \( 1 - 2.27iT - 47T^{2} \) |
| 53 | \( 1 - 9.84T + 53T^{2} \) |
| 59 | \( 1 + 5.87T + 59T^{2} \) |
| 61 | \( 1 + 7.02iT - 61T^{2} \) |
| 67 | \( 1 + 10.8iT - 67T^{2} \) |
| 71 | \( 1 + 0.0512iT - 71T^{2} \) |
| 73 | \( 1 - 2.86T + 73T^{2} \) |
| 79 | \( 1 - 7.00T + 79T^{2} \) |
| 83 | \( 1 + 7.59iT - 83T^{2} \) |
| 89 | \( 1 - 9.72T + 89T^{2} \) |
| 97 | \( 1 + 1.77T + 97T^{2} \) |
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
−10.62195505501770349005937069721, −9.303153354045109551602197835138, −8.735684551253562846387036842804, −7.78863221288263393999106538518, −6.58948173773911539629128432122, −5.95521519001421043759284711035, −4.64743779800719060651801197273, −3.82530933738351017127731350330, −3.26126883567804106209310218561, −0.77519926117922354515388047905,
0.862509350061340601957447676348, 2.60889436895321989979620317210, 3.55493191309172997624333697747, 4.96239706872142362646694062452, 5.90965394256081618818318492484, 6.86169652119606730291407490739, 7.37944503983509008168096250592, 8.454232352197533363611070245639, 9.075951381025776841970265223563, 10.32832378265692564563790490742