L(s) = 1 | + (−0.239 − 1.71i)3-s + (1.06 + 1.06i)5-s + (−0.366 + 1.36i)7-s + (−2.88 + 0.820i)9-s + (1.06 + 3.97i)11-s + (3.59 − 0.232i)13-s + (1.57 − 2.08i)15-s + (2.51 + 4.36i)17-s + (3.73 + i)19-s + (2.43 + 0.301i)21-s − 2.73i·25-s + (2.09 + 4.75i)27-s + (−6.20 − 3.58i)29-s + (2.46 − 2.46i)31-s + (6.56 − 2.77i)33-s + ⋯ |
L(s) = 1 | + (−0.138 − 0.990i)3-s + (0.476 + 0.476i)5-s + (−0.138 + 0.516i)7-s + (−0.961 + 0.273i)9-s + (0.321 + 1.19i)11-s + (0.997 − 0.0643i)13-s + (0.405 − 0.537i)15-s + (0.611 + 1.05i)17-s + (0.856 + 0.229i)19-s + (0.530 + 0.0657i)21-s − 0.546i·25-s + (0.403 + 0.914i)27-s + (−1.15 − 0.665i)29-s + (0.442 − 0.442i)31-s + (1.14 − 0.483i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.52949 + 0.125959i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.52949 + 0.125959i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.239 + 1.71i)T \) |
| 13 | \( 1 + (-3.59 + 0.232i)T \) |
good | 5 | \( 1 + (-1.06 - 1.06i)T + 5iT^{2} \) |
| 7 | \( 1 + (0.366 - 1.36i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-1.06 - 3.97i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-2.51 - 4.36i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.73 - i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (6.20 + 3.58i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.46 + 2.46i)T - 31iT^{2} \) |
| 37 | \( 1 + (5.23 - 1.40i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-5.42 + 1.45i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (1.90 - 1.09i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.25 + 4.25i)T - 47iT^{2} \) |
| 53 | \( 1 - 0.779iT - 53T^{2} \) |
| 59 | \( 1 + (-2.90 - 0.779i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.53 - 5.73i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-0.779 + 2.90i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (0.901 + 0.901i)T + 73iT^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 + (2.90 + 2.90i)T + 83iT^{2} \) |
| 89 | \( 1 + (2.41 + 9.01i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-1.63 - 0.437i)T + (84.0 + 48.5i)T^{2} \) |
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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
−10.63162505732510799705912476867, −9.809655592383935134690375621540, −8.809555355049436007035222801552, −7.87912782972042850583559908788, −7.03021847301314523588801292371, −6.11231154130988006972898389438, −5.55122816303582697191429348706, −3.89607773803268357450881601478, −2.52906036445032653005205020099, −1.49650738571084111305570000705,
0.990322527641059224456056828872, 3.13402229082820806357964382122, 3.87435732304216713120584090101, 5.22725742127030344698208784566, 5.72122261263827855119112108299, 6.92759666815694778927253645403, 8.202426856638977999080657739374, 9.135781390681919862073263218390, 9.534956327310378269628503883415, 10.67597410869572919082642876347