| L(s) = 1 | − 1.73i·3-s + (−2 − 1.73i)7-s − 2.99·9-s − 6.92i·13-s + 3.46i·19-s + (−2.99 + 3.46i)21-s − 5·25-s + 5.19i·27-s − 10.3i·31-s + 10·37-s − 11.9·39-s + 8·43-s + (1.00 + 6.92i)49-s + 5.99·57-s + 6.92i·61-s + ⋯ |
| L(s) = 1 | − 0.999i·3-s + (−0.755 − 0.654i)7-s − 0.999·9-s − 1.92i·13-s + 0.794i·19-s + (−0.654 + 0.755i)21-s − 25-s + 0.999i·27-s − 1.86i·31-s + 1.64·37-s − 1.92·39-s + 1.21·43-s + (0.142 + 0.989i)49-s + 0.794·57-s + 0.887i·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.408275 - 0.893674i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.408275 - 0.893674i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 6.92iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 10.3iT - 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 16T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 13.8iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 13.8iT - 97T^{2} \) |
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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
−11.27016985117609952275276913441, −10.32085686862909138816537058895, −9.426005045823869146915821811492, −7.906473234830046661920355803983, −7.68377899554010626929577658630, −6.28416576153320178576308488200, −5.64076300145299839501407852730, −3.81534610610108545815657166518, −2.59560345805036421184312942000, −0.68172367498828410801151643175,
2.45998982429343198460841872016, 3.78765543788006990020417088184, 4.80677101815720239466481914720, 6.00062932724666866168488079322, 6.92201393877661151920413411869, 8.494320534823612833951768959269, 9.299738433359515655363326654130, 9.775210639448240036392046645199, 11.05144816216089049225488869585, 11.68422357551031311754188757406
