L(s) = 1 | + (1.67 + 1.48i)5-s − i·7-s + 1.67·11-s − 0.869i·13-s + 1.86i·17-s − 0.869·19-s + i·23-s + (0.612 + 4.96i)25-s + 2.44·29-s − 4.19·31-s + (1.48 − 1.67i)35-s + 2.76i·37-s + 10.5·41-s + 7.11i·43-s + 2.71i·47-s + ⋯ |
L(s) = 1 | + (0.749 + 0.662i)5-s − 0.377i·7-s + 0.505·11-s − 0.241i·13-s + 0.453i·17-s − 0.199·19-s + 0.208i·23-s + (0.122 + 0.992i)25-s + 0.453·29-s − 0.753·31-s + (0.250 − 0.283i)35-s + 0.455i·37-s + 1.65·41-s + 1.08i·43-s + 0.395i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.237824706\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.237824706\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.67 - 1.48i)T \) |
| 23 | \( 1 - iT \) |
good | 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 - 1.67T + 11T^{2} \) |
| 13 | \( 1 + 0.869iT - 13T^{2} \) |
| 17 | \( 1 - 1.86iT - 17T^{2} \) |
| 19 | \( 1 + 0.869T + 19T^{2} \) |
| 29 | \( 1 - 2.44T + 29T^{2} \) |
| 31 | \( 1 + 4.19T + 31T^{2} \) |
| 37 | \( 1 - 2.76iT - 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 - 7.11iT - 43T^{2} \) |
| 47 | \( 1 - 2.71iT - 47T^{2} \) |
| 53 | \( 1 + 3.63iT - 53T^{2} \) |
| 59 | \( 1 - 3.48T + 59T^{2} \) |
| 61 | \( 1 + 1.78T + 61T^{2} \) |
| 67 | \( 1 - 11.0iT - 67T^{2} \) |
| 71 | \( 1 - 0.100T + 71T^{2} \) |
| 73 | \( 1 + 2.16iT - 73T^{2} \) |
| 79 | \( 1 + 5.08T + 79T^{2} \) |
| 83 | \( 1 + 9.09iT - 83T^{2} \) |
| 89 | \( 1 - 7.92T + 89T^{2} \) |
| 97 | \( 1 + 4.99iT - 97T^{2} \) |
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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
−8.586601277475668062830874433436, −7.63122650800431891231367749426, −7.07166840956230366047415339491, −6.22046902816896442285243394492, −5.79692498873260201926838122127, −4.74799440751110947052181080102, −3.88217309326386503145585139032, −3.03806256269962482619049392390, −2.11868467680904447543220105861, −1.08899862761336846063769993475,
0.71826951436879564340688938387, 1.86425283459147378013835089034, 2.61524230925507835667382131899, 3.82796170849969458499266567742, 4.60661614660501599214730227661, 5.42661653450323545776899710131, 6.00713896534416589169070425723, 6.78960351360965302223866785076, 7.58799423136871716191759659198, 8.530979830526883460809187692814