| L(s) = 1 | + 5.65i·5-s + 1.41i·11-s − 16·13-s − 16.9i·17-s + 8·19-s − 12.7i·23-s − 7.00·25-s − 24.0i·29-s − 56·31-s + 24·37-s − 50.9i·41-s + 40·43-s − 11.3i·47-s + 43.8i·53-s − 8.00·55-s + ⋯ |
| L(s) = 1 | + 1.13i·5-s + 0.128i·11-s − 1.23·13-s − 0.998i·17-s + 0.421·19-s − 0.553i·23-s − 0.280·25-s − 0.829i·29-s − 1.80·31-s + 0.648·37-s − 1.24i·41-s + 0.930·43-s − 0.240i·47-s + 0.827i·53-s − 0.145·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.264878805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.264878805\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 5.65iT - 25T^{2} \) |
| 11 | \( 1 - 1.41iT - 121T^{2} \) |
| 13 | \( 1 + 16T + 169T^{2} \) |
| 17 | \( 1 + 16.9iT - 289T^{2} \) |
| 19 | \( 1 - 8T + 361T^{2} \) |
| 23 | \( 1 + 12.7iT - 529T^{2} \) |
| 29 | \( 1 + 24.0iT - 841T^{2} \) |
| 31 | \( 1 + 56T + 961T^{2} \) |
| 37 | \( 1 - 24T + 1.36e3T^{2} \) |
| 41 | \( 1 + 50.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 40T + 1.84e3T^{2} \) |
| 47 | \( 1 + 11.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 43.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 11.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 40T + 3.72e3T^{2} \) |
| 67 | \( 1 + 26T + 4.48e3T^{2} \) |
| 71 | \( 1 + 134. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 88T + 5.32e3T^{2} \) |
| 79 | \( 1 - 82T + 6.24e3T^{2} \) |
| 83 | \( 1 - 101. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 62.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 40T + 9.40e3T^{2} \) |
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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
−9.299572500740108035584039980730, −7.986568223184262002495176903109, −7.26819349846996756260529573336, −6.84942102698993693741379826829, −5.76831793517918835632354930526, −4.93992726360416959825230377464, −3.88319072719304708276180513100, −2.83576671867718417652802638264, −2.18228056529006350423147168805, −0.37509649299992521882235774428,
1.00948795145379293645583423982, 2.07531606245200864637350655803, 3.37424733066635923021837648523, 4.38178248102862171392064957625, 5.16694053316821937449359126387, 5.79916892939248322149706219282, 6.96897813714311794120400052864, 7.73038998452923305473566283611, 8.482641828824218872407911048008, 9.267497239575672460544332188642
