L(s) = 1 | + (−0.0440 + 0.0254i)5-s + (4.52 − 7.84i)7-s + (3.29 + 1.90i)11-s + (−0.216 − 0.375i)13-s − 26.2i·17-s − 34.2·19-s + (29.9 − 17.3i)23-s + (−12.4 + 21.6i)25-s + (14.0 + 8.10i)29-s + (−17.1 − 29.7i)31-s + 0.460i·35-s − 29.2·37-s + (−48.7 + 28.1i)41-s + (3.94 − 6.83i)43-s + (33.4 + 19.3i)47-s + ⋯ |
L(s) = 1 | + (−0.00880 + 0.00508i)5-s + (0.647 − 1.12i)7-s + (0.299 + 0.172i)11-s + (−0.0166 − 0.0288i)13-s − 1.54i·17-s − 1.80·19-s + (1.30 − 0.752i)23-s + (−0.499 + 0.865i)25-s + (0.483 + 0.279i)29-s + (−0.553 − 0.959i)31-s + 0.0131i·35-s − 0.791·37-s + (−1.18 + 0.685i)41-s + (0.0917 − 0.158i)43-s + (0.711 + 0.410i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.797 + 0.603i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.797 + 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.218953288\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.218953288\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.0440 - 0.0254i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-4.52 + 7.84i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-3.29 - 1.90i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (0.216 + 0.375i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 26.2iT - 289T^{2} \) |
| 19 | \( 1 + 34.2T + 361T^{2} \) |
| 23 | \( 1 + (-29.9 + 17.3i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-14.0 - 8.10i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (17.1 + 29.7i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 29.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + (48.7 - 28.1i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-3.94 + 6.83i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-33.4 - 19.3i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 50.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (8.54 - 4.93i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-36.5 + 63.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-12.6 - 21.9i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 97.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 77.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + (42.1 - 72.9i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-40.6 - 23.4i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 108. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (32.4 - 56.1i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−8.785426200915791112593954403012, −8.034808192061363607773233614604, −7.04714688700478806185160263601, −6.77680365969697090530208531469, −5.40124523774195477602970752764, −4.61459086471149407904981275956, −3.91930639604653285888090696917, −2.70926049682672661201191273370, −1.50088489684653561499663480315, −0.31246093977749412606350861851,
1.53562694455455529686181960881, 2.34096427472597273601036494649, 3.58684635544399245700670122144, 4.52412863985584897181137879430, 5.46134722302576186429269367808, 6.17022057413553854436485897379, 6.97408061766886225302760043052, 8.169299408686429669393125065685, 8.629617579517942014724374126454, 9.157587942949814197656626209649