| L(s) = 1 | + 3.32·5-s + 4.21·7-s − 16.0·11-s − 17.6i·13-s − 4.44i·17-s + 26.5i·19-s − 5.59i·23-s − 13.9·25-s − 50.0·29-s − 11.4·31-s + 14.0·35-s + 24.5i·37-s + 18.5i·41-s − 37.2i·43-s + 73.5i·47-s + ⋯ |
| L(s) = 1 | + 0.664·5-s + 0.602·7-s − 1.46·11-s − 1.35i·13-s − 0.261i·17-s + 1.39i·19-s − 0.243i·23-s − 0.558·25-s − 1.72·29-s − 0.370·31-s + 0.400·35-s + 0.664i·37-s + 0.453i·41-s − 0.865i·43-s + 1.56i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1455374508\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1455374508\) |
| \(L(2)\) |
|
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 | \( 1 - 3.32T + 25T^{2} \) |
| 7 | \( 1 - 4.21T + 49T^{2} \) |
| 11 | \( 1 + 16.0T + 121T^{2} \) |
| 13 | \( 1 + 17.6iT - 169T^{2} \) |
| 17 | \( 1 + 4.44iT - 289T^{2} \) |
| 19 | \( 1 - 26.5iT - 361T^{2} \) |
| 23 | \( 1 + 5.59iT - 529T^{2} \) |
| 29 | \( 1 + 50.0T + 841T^{2} \) |
| 31 | \( 1 + 11.4T + 961T^{2} \) |
| 37 | \( 1 - 24.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 18.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 37.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 73.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 2.32T + 2.80e3T^{2} \) |
| 59 | \( 1 + 26.8T + 3.48e3T^{2} \) |
| 61 | \( 1 - 12.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 45.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 83.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 3.04T + 5.32e3T^{2} \) |
| 79 | \( 1 - 17.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 92.8T + 6.88e3T^{2} \) |
| 89 | \( 1 + 160. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 122.T + 9.40e3T^{2} \) |
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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
−9.716442750726231691184083135297, −8.555761280422977886819137110160, −7.86163203921063060685409885029, −7.41392541837042417282724406777, −5.90069919801130685736994438111, −5.61887931716684941989612961314, −4.74907595328306988526067649625, −3.48789693405533379257375418260, −2.51317643574273128510675171487, −1.51563065348854720504616119486,
0.03422988169136405301843333513, 1.77555321992238272022652406534, 2.38491292336841358265609066943, 3.72579339428961668911040702533, 4.82660654246781167198934076006, 5.39026150582279451558176048160, 6.33016052766320404291177741361, 7.28093545379080509913081904672, 7.890683061730221508596857310266, 8.953029472249098688587636256962
