L(s) = 1 | + i·2-s − 4-s − i·7-s − i·8-s − i·9-s + i·11-s + 14-s + 16-s + 18-s − 22-s + 2·23-s − i·25-s + i·28-s + (1 − i)29-s + i·32-s + ⋯ |
L(s) = 1 | + i·2-s − 4-s − i·7-s − i·8-s − i·9-s + i·11-s + 14-s + 16-s + 18-s − 22-s + 2·23-s − i·25-s + i·28-s + (1 − i)29-s + i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9558381448\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9558381448\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 - iT \) |
good | 3 | \( 1 + iT^{2} \) |
| 5 | \( 1 + iT^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 - 2T + T^{2} \) |
| 29 | \( 1 + (-1 + i)T - iT^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (-1 + i)T - iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + (1 - i)T - iT^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (1 - i)T - iT^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + iT^{2} \) |
| 67 | \( 1 + (1 - i)T - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.760830129047799293149570581009, −9.155355789924776686178107903898, −8.181228497933910156839770416106, −7.33670030539146645655917193253, −6.75816045040694582501176208116, −6.03835119857398108944494914105, −4.70868366702665767374422414198, −4.26574191193496485290840428212, −3.06763672112963681062752762687, −0.989610555139790871486931856961,
1.46627190504452614334138868886, 2.73509773527084885760579198521, 3.34372258427751805232027782387, 4.93475883510108462209291864028, 5.21065979826167867590711182093, 6.39539641704811143769136122421, 7.69130206206045447274801625390, 8.613333122023132934311182172769, 8.957368339053926846002449301592, 9.961170035865710867756487488410