| L(s) = 1 | + (−4.89 − 2.82i)5-s + (−3 − 5.19i)7-s + (−4.89 + 2.82i)11-s + (−5 + 8.66i)13-s − 22.6i·17-s − 2·19-s + (−9.79 − 5.65i)23-s + (3.49 + 6.06i)25-s + (14.6 − 8.48i)29-s + (−11 + 19.0i)31-s + 33.9i·35-s − 6·37-s + (−29.3 − 16.9i)41-s + (41 + 71.0i)43-s + (−58.7 + 33.9i)47-s + ⋯ |
| L(s) = 1 | + (−0.979 − 0.565i)5-s + (−0.428 − 0.742i)7-s + (−0.445 + 0.257i)11-s + (−0.384 + 0.666i)13-s − 1.33i·17-s − 0.105·19-s + (−0.425 − 0.245i)23-s + (0.139 + 0.242i)25-s + (0.506 − 0.292i)29-s + (−0.354 + 0.614i)31-s + 0.969i·35-s − 0.162·37-s + (−0.716 − 0.413i)41-s + (0.953 + 1.65i)43-s + (−1.25 + 0.722i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5626583917\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5626583917\) |
| \(L(2)\) |
|
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 | \( 1 + (4.89 + 2.82i)T + (12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (3 + 5.19i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (4.89 - 2.82i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (5 - 8.66i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 22.6iT - 289T^{2} \) |
| 19 | \( 1 + 2T + 361T^{2} \) |
| 23 | \( 1 + (9.79 + 5.65i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-14.6 + 8.48i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (11 - 19.0i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 6T + 1.36e3T^{2} \) |
| 41 | \( 1 + (29.3 + 16.9i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-41 - 71.0i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (58.7 - 33.9i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 62.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-63.6 - 36.7i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-43 - 74.4i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-1 + 1.73i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 124. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 82T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-5 - 8.66i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-63.6 + 36.7i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 33.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-47 - 81.4i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−9.685543350527030923730828377336, −8.755686194115794639558042186591, −7.936668268929068127905825566356, −7.23095170991221867126950891206, −6.55603416426663239608331510380, −5.16331047688746938641316867344, −4.47774312350870091013605927765, −3.67180256966502655569394749269, −2.48195252985312243735948945571, −0.850097899783511490600703924532,
0.21535025259206657182832328526, 2.11547532501127587789490625700, 3.21631563637121823520533680955, 3.87569021277601549799619915152, 5.16792823356867850610204132481, 5.99181754824136734800284811223, 6.86137980000752916600383386193, 7.84666862941873213035016376646, 8.291309998034736327129457708237, 9.275182777183047418858970713042
