L(s) = 1 | + 1.41i·2-s − 2.00·4-s + (−0.707 − 0.707i)5-s − 3i·7-s − 2.82i·8-s + (1.00 − 1.00i)10-s + (3.53 + 3.53i)11-s + (1 − i)13-s + 4.24·14-s + 4.00·16-s + 4.24·17-s + (4 − 4i)19-s + (1.41 + 1.41i)20-s + (−5.00 + 5.00i)22-s + 2.82i·23-s + ⋯ |
L(s) = 1 | + 0.999i·2-s − 1.00·4-s + (−0.316 − 0.316i)5-s − 1.13i·7-s − 1.00i·8-s + (0.316 − 0.316i)10-s + (1.06 + 1.06i)11-s + (0.277 − 0.277i)13-s + 1.13·14-s + 1.00·16-s + 1.02·17-s + (0.917 − 0.917i)19-s + (0.316 + 0.316i)20-s + (−1.06 + 1.06i)22-s + 0.589i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.23158 + 0.244976i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23158 + 0.244976i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.707 + 0.707i)T + 5iT^{2} \) |
| 7 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 + (-3.53 - 3.53i)T + 11iT^{2} \) |
| 13 | \( 1 + (-1 + i)T - 13iT^{2} \) |
| 17 | \( 1 - 4.24T + 17T^{2} \) |
| 19 | \( 1 + (-4 + 4i)T - 19iT^{2} \) |
| 23 | \( 1 - 2.82iT - 23T^{2} \) |
| 29 | \( 1 + (-2.82 + 2.82i)T - 29iT^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 + (2 + 2i)T + 37iT^{2} \) |
| 41 | \( 1 + 5.65iT - 41T^{2} \) |
| 43 | \( 1 + (-7 - 7i)T + 43iT^{2} \) |
| 47 | \( 1 + 4.24T + 47T^{2} \) |
| 53 | \( 1 + (0.707 + 0.707i)T + 53iT^{2} \) |
| 59 | \( 1 + (7.07 + 7.07i)T + 59iT^{2} \) |
| 61 | \( 1 + (-10 + 10i)T - 61iT^{2} \) |
| 67 | \( 1 + (-1 + i)T - 67iT^{2} \) |
| 71 | \( 1 - 15.5iT - 71T^{2} \) |
| 73 | \( 1 - 15iT - 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 + (7.77 - 7.77i)T - 83iT^{2} \) |
| 89 | \( 1 + 1.41iT - 89T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
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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
−11.19576452969721465652139468441, −9.931005045558722894783029401260, −9.436784910739632970107072832850, −8.238299102829853821791758475115, −7.36904087970834219375644990118, −6.81752404760477237939586679878, −5.50197049362225323294404543938, −4.42636901150732962099004719497, −3.65042521964223451480747970573, −0.999190478304177383108947652588,
1.42832791712060923972908294814, 3.04983304176838091241052994574, 3.76483011059339621029102631114, 5.31126208466626089564556919339, 6.10129139749155499702450030233, 7.65763996633432484377989348785, 8.753116351751879270384504967628, 9.217444719297262353764642623054, 10.33844210618699093171829325537, 11.27505313208376439426094580464