L(s) = 1 | + (2 − 2i)3-s − 5i·9-s + (−2 − 2i)11-s − 6·17-s + (6 − 6i)19-s − 5i·25-s + (−4 − 4i)27-s − 8·33-s − 6i·41-s + (6 + 6i)43-s + 7·49-s + (−12 + 12i)51-s − 24i·57-s + (10 + 10i)59-s + (−6 + 6i)67-s + ⋯ |
L(s) = 1 | + (1.15 − 1.15i)3-s − 1.66i·9-s + (−0.603 − 0.603i)11-s − 1.45·17-s + (1.37 − 1.37i)19-s − i·25-s + (−0.769 − 0.769i)27-s − 1.39·33-s − 0.937i·41-s + (0.914 + 0.914i)43-s + 49-s + (−1.68 + 1.68i)51-s − 3.17i·57-s + (1.30 + 1.30i)59-s + (−0.733 + 0.733i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.118156723\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.118156723\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (-2 + 2i)T - 3iT^{2} \) |
| 5 | \( 1 + 5iT^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + (2 + 2i)T + 11iT^{2} \) |
| 13 | \( 1 - 13iT^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + (-6 + 6i)T - 19iT^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 29iT^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37iT^{2} \) |
| 41 | \( 1 + 6iT - 41T^{2} \) |
| 43 | \( 1 + (-6 - 6i)T + 43iT^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 + (-10 - 10i)T + 59iT^{2} \) |
| 61 | \( 1 - 61iT^{2} \) |
| 67 | \( 1 + (6 - 6i)T - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (2 - 2i)T - 83iT^{2} \) |
| 89 | \( 1 - 18iT - 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−9.284251650252961596063019202131, −8.803849558941571265239170047156, −8.020284198885914417639769930380, −7.23827562527942798807709696167, −6.62717317503173937190731280993, −5.47715606407333095782087229301, −4.20231806608114379710959866625, −2.88459125440640351331937796670, −2.36445753953027619438674644962, −0.834582311387276827991493732713,
1.98773776291656032079881168238, 3.05371656727296707824483995781, 3.92442608162171711703254581100, 4.77111021245306703550164216763, 5.67865443344435722311545911789, 7.11827242659521147135499863243, 7.88182023823245553295176695194, 8.698799544335503279115229005473, 9.434506558798499245890033369067, 10.00167753879394501391472893078