| L(s) = 1 | + 0.517·2-s − 1.73·4-s + (2.44 + i)7-s − 1.93·8-s + 0.378i·11-s − 1.79·13-s + (1.26 + 0.517i)14-s + 2.46·16-s + 3.46i·17-s − 1.79i·19-s + 0.196i·22-s − 1.41·23-s − 0.928·26-s + (−4.24 − 1.73i)28-s + 1.41i·29-s + ⋯ |
| L(s) = 1 | + 0.366·2-s − 0.866·4-s + (0.925 + 0.377i)7-s − 0.683·8-s + 0.114i·11-s − 0.497·13-s + (0.338 + 0.138i)14-s + 0.616·16-s + 0.840i·17-s − 0.411i·19-s + 0.0418i·22-s − 0.294·23-s − 0.182·26-s + (−0.801 − 0.327i)28-s + 0.262i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.103 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.103 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.254165773\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.254165773\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.44 - i)T \) |
| good | 2 | \( 1 - 0.517T + 2T^{2} \) |
| 11 | \( 1 - 0.378iT - 11T^{2} \) |
| 13 | \( 1 + 1.79T + 13T^{2} \) |
| 17 | \( 1 - 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 1.79iT - 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 6.69iT - 31T^{2} \) |
| 37 | \( 1 + 1.46iT - 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 4.92iT - 43T^{2} \) |
| 47 | \( 1 - 9.46iT - 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 9.46T + 59T^{2} \) |
| 61 | \( 1 - 13.3iT - 61T^{2} \) |
| 67 | \( 1 + 10.9iT - 67T^{2} \) |
| 71 | \( 1 - 15.0iT - 71T^{2} \) |
| 73 | \( 1 - 1.79T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 9.46iT - 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−9.556822223378013487145753148165, −8.722238927473774790833834574675, −8.251630419507556677781700727719, −7.31299237926590075816063663927, −6.21147051939665329174683519327, −5.30927938209780805531164090585, −4.73436083973583668049635014131, −3.87987183007340125560306949824, −2.73273487372070050737218981951, −1.39514354038301288256700190364,
0.46636028388088369270718040125, 2.01740351058428947647024926190, 3.37829068087012951650351890504, 4.26265821298468803604117439132, 5.00788626494694493021413290386, 5.64843186750317258901592044932, 6.81653151973044259927222164126, 7.76546569468002580198232893597, 8.354532132846676421332057698097, 9.220077603122869294705352777642
