L(s) = 1 | + (−2.80 + 2.80i)2-s − 7.70i·4-s + (25.0 + 25.0i)7-s + (−0.817 − 0.817i)8-s − 42.1i·11-s + (25.7 − 25.7i)13-s − 140.·14-s + 66.2·16-s + (42.0 − 42.0i)17-s − 58.9i·19-s + (118. + 118. i)22-s + (141. + 141. i)23-s + 144. i·26-s + (192. − 192. i)28-s − 79.0·29-s + ⋯ |
L(s) = 1 | + (−0.990 + 0.990i)2-s − 0.963i·4-s + (1.35 + 1.35i)7-s + (−0.0361 − 0.0361i)8-s − 1.15i·11-s + (0.548 − 0.548i)13-s − 2.67·14-s + 1.03·16-s + (0.600 − 0.600i)17-s − 0.711i·19-s + (1.14 + 1.14i)22-s + (1.28 + 1.28i)23-s + 1.08i·26-s + (1.30 − 1.30i)28-s − 0.506·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.161 - 0.986i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.943076 + 0.800984i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.943076 + 0.800984i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (2.80 - 2.80i)T - 8iT^{2} \) |
| 7 | \( 1 + (-25.0 - 25.0i)T + 343iT^{2} \) |
| 11 | \( 1 + 42.1iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (-25.7 + 25.7i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 + (-42.0 + 42.0i)T - 4.91e3iT^{2} \) |
| 19 | \( 1 + 58.9iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-141. - 141. i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 + 79.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 184.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (81.4 + 81.4i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 + 14.0iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (152. - 152. i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + (199. - 199. i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 + (-84.5 - 84.5i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 - 665.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 600.T + 2.26e5T^{2} \) |
| 67 | \( 1 + (6.22 + 6.22i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 - 750. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (418. - 418. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 + 825. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-463. - 463. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 + 646.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (371. + 371. i)T + 9.12e5iT^{2} \) |
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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
−11.66573606677671227364541024849, −11.12644162080495518591429169758, −9.647687558880881278328208667568, −8.676343152048609232206496760274, −8.291579399487282272297903708183, −7.23500819453145418212867245578, −5.84923670127124264386585072715, −5.23093828089027728505287135056, −3.04560913656835136674617235692, −1.06409702849842237315292946078,
1.01244447409633453696285146627, 1.95037274308931248816896109910, 3.81234908709937666494269512091, 4.96502540091155295024923009281, 6.84568341282738302407655277491, 7.934587786849536753548312614814, 8.656918497398585076380499472861, 10.01574898708577060898505100890, 10.48004433682606613571122971133, 11.33832879237366148474058458543