L(s) = 1 | + (−1 − i)2-s + 3-s + i·4-s + (−1 − i)6-s + 9-s + i·12-s − 2i·13-s + 16-s + (−1 − i)18-s + i·23-s + 25-s + (−2 + 2i)26-s + 27-s − i·29-s + (−1 + i)31-s + (−1 − i)32-s + ⋯ |
L(s) = 1 | + (−1 − i)2-s + 3-s + i·4-s + (−1 − i)6-s + 9-s + i·12-s − 2i·13-s + 16-s + (−1 − i)18-s + i·23-s + 25-s + (−2 + 2i)26-s + 27-s − i·29-s + (−1 + i)31-s + (−1 − i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9781764484\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9781764484\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 23 | \( 1 - iT \) |
| 29 | \( 1 + iT \) |
good | 2 | \( 1 + (1 + i)T + iT^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 13 | \( 1 + 2iT - T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 31 | \( 1 + (1 - i)T - iT^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + (-1 + i)T - iT^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (1 - i)T - iT^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + 2iT - T^{2} \) |
| 61 | \( 1 + iT^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1 + i)T + iT^{2} \) |
| 79 | \( 1 + iT^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - iT^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.226196205317528357531239238560, −8.512134568620150899138281541910, −7.87582107007459813947343431808, −7.31186983166383381577428667384, −5.93340839207471549138623551794, −4.99828753289512800788046547803, −3.57422067079487566364897976209, −3.05820833930307492662366687208, −2.13755445319794556580613859946, −0.986810074274520295692088188142,
1.44824485471940837303575561174, 2.63311178906281489371066347836, 3.85595751568312228144040158792, 4.65406684250847171309839188439, 5.98407845037203523055299848919, 6.95618804459985683326613097557, 7.11460933333027645543764048647, 8.133882242002105114955438928583, 8.890001531197521807206290837304, 9.100987847803824006949055037011