L(s) = 1 | + (1.38 + 0.297i)2-s + (1.82 + 0.822i)4-s + 3.36i·5-s − 2.64·7-s + (2.27 + 1.68i)8-s + (−1 + 4.64i)10-s − 2.16i·11-s − 4.64i·13-s + (−3.65 − 0.787i)14-s + (2.64 + 3i)16-s + 4.55·17-s − 6.29i·19-s + (−2.76 + 6.12i)20-s + (0.645 − 3i)22-s + 0.979·23-s + ⋯ |
L(s) = 1 | + (0.977 + 0.210i)2-s + (0.911 + 0.411i)4-s + 1.50i·5-s − 0.999·7-s + (0.804 + 0.594i)8-s + (−0.316 + 1.46i)10-s − 0.654i·11-s − 1.28i·13-s + (−0.977 − 0.210i)14-s + (0.661 + 0.750i)16-s + 1.10·17-s − 1.44i·19-s + (−0.618 + 1.36i)20-s + (0.137 − 0.639i)22-s + 0.204·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.594 - 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.594 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.77414 + 0.895352i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.77414 + 0.895352i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.38 - 0.297i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 3.36iT - 5T^{2} \) |
| 7 | \( 1 + 2.64T + 7T^{2} \) |
| 11 | \( 1 + 2.16iT - 11T^{2} \) |
| 13 | \( 1 + 4.64iT - 13T^{2} \) |
| 17 | \( 1 - 4.55T + 17T^{2} \) |
| 19 | \( 1 + 6.29iT - 19T^{2} \) |
| 23 | \( 1 - 0.979T + 23T^{2} \) |
| 29 | \( 1 - 4.33iT - 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 1.35iT - 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 - 3.29iT - 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 - 4.33iT - 53T^{2} \) |
| 59 | \( 1 + 11.2iT - 59T^{2} \) |
| 61 | \( 1 - 1.93iT - 61T^{2} \) |
| 67 | \( 1 - 3iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 + 8.64T + 79T^{2} \) |
| 83 | \( 1 - 2.38iT - 83T^{2} \) |
| 89 | \( 1 + 4.55T + 89T^{2} \) |
| 97 | \( 1 - 2.29T + 97T^{2} \) |
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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
−12.68759965421486003488660147393, −11.49488998552697732771078915411, −10.71468009002761886938035028474, −9.902944541187599022967127698884, −8.139272618576062648627200331292, −7.01155521538036219663284778642, −6.37044472035922622085659203509, −5.26014428592131849951658295557, −3.26223376700553137371537424935, −3.02544171354141731788805282860,
1.66349342426509532898164224007, 3.61570499855101560180919175187, 4.65207255136268861503313462328, 5.71426751470152408577774003751, 6.80466890639347787487416176111, 8.142443998723014410803790011523, 9.518310897213346539207455005712, 10.09063842505531341956138723830, 11.80282865511744937508578045947, 12.24098562866879298102831300458