L(s) = 1 | + (2.61e5 + 2.22e4i)2-s + 2.96e8i·3-s + (6.77e10 + 1.16e10i)4-s + 1.46e12·5-s + (−6.58e12 + 7.73e13i)6-s − 2.38e15i·7-s + (1.74e16 + 4.53e15i)8-s + 6.23e16·9-s + (3.82e17 + 3.25e16i)10-s − 1.01e19i·11-s + (−3.43e18 + 2.00e19i)12-s − 4.81e19·13-s + (5.30e19 − 6.23e20i)14-s + 4.33e20i·15-s + (4.45e21 + 1.57e21i)16-s + 1.19e22·17-s + ⋯ |
L(s) = 1 | + (0.996 + 0.0847i)2-s + 0.764i·3-s + (0.985 + 0.168i)4-s + 0.383·5-s + (−0.0648 + 0.761i)6-s − 1.46i·7-s + (0.967 + 0.251i)8-s + 0.415·9-s + (0.382 + 0.0325i)10-s − 1.81i·11-s + (−0.129 + 0.753i)12-s − 0.427·13-s + (0.124 − 1.46i)14-s + 0.293i·15-s + (0.942 + 0.333i)16-s + 0.850·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.168i)\, \overline{\Lambda}(37-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+18) \, L(s)\cr =\mathstrut & (0.985 + 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{37}{2})\) |
\(\approx\) |
\(4.489267042\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.489267042\) |
\(L(19)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.61e5 - 2.22e4i)T \) |
good | 3 | \( 1 - 2.96e8iT - 1.50e17T^{2} \) |
| 5 | \( 1 - 1.46e12T + 1.45e25T^{2} \) |
| 7 | \( 1 + 2.38e15iT - 2.65e30T^{2} \) |
| 11 | \( 1 + 1.01e19iT - 3.09e37T^{2} \) |
| 13 | \( 1 + 4.81e19T + 1.26e40T^{2} \) |
| 17 | \( 1 - 1.19e22T + 1.97e44T^{2} \) |
| 19 | \( 1 + 1.66e22iT - 1.08e46T^{2} \) |
| 23 | \( 1 - 2.63e24iT - 1.05e49T^{2} \) |
| 29 | \( 1 - 1.36e26T + 4.42e52T^{2} \) |
| 31 | \( 1 + 4.52e26iT - 4.88e53T^{2} \) |
| 37 | \( 1 - 5.31e27T + 2.85e56T^{2} \) |
| 41 | \( 1 - 1.73e29T + 1.14e58T^{2} \) |
| 43 | \( 1 - 2.26e29iT - 6.38e58T^{2} \) |
| 47 | \( 1 + 1.08e30iT - 1.56e60T^{2} \) |
| 53 | \( 1 + 1.23e31T + 1.18e62T^{2} \) |
| 59 | \( 1 - 9.54e31iT - 5.63e63T^{2} \) |
| 61 | \( 1 + 5.38e31T + 1.87e64T^{2} \) |
| 67 | \( 1 + 6.26e32iT - 5.47e65T^{2} \) |
| 71 | \( 1 - 2.83e33iT - 4.41e66T^{2} \) |
| 73 | \( 1 + 6.34e33T + 1.20e67T^{2} \) |
| 79 | \( 1 - 2.21e34iT - 2.06e68T^{2} \) |
| 83 | \( 1 + 6.15e34iT - 1.22e69T^{2} \) |
| 89 | \( 1 + 8.78e34T + 1.50e70T^{2} \) |
| 97 | \( 1 - 1.81e35T + 3.34e71T^{2} \) |
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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
−16.00215723937804429577959467180, −14.25990980299999720603577761408, −13.30028666865640048720277312798, −11.19965280015531061816180123152, −10.04104841813583769191198408959, −7.55417062322113787388420231550, −5.84629944555684351116745461542, −4.30273132739199656197371463377, −3.25973076986313208690850435046, −1.06896055379920513725414306427,
1.66782833515984445829227209120, 2.49806010945054053509721309488, 4.70394974500679176404194006159, 6.12582981616502223385106950463, 7.50177532050709937022560789708, 9.869518858813929121769505578823, 12.19197052102525829641922877602, 12.62451949054900662122080757705, 14.46200044080201246490929633427, 15.65780241003025340486809186645