L(s) = 1 | + (1.36 + 0.366i)2-s + (0.578 + 2.15i)3-s + (2.15 + 0.578i)5-s + 3.16i·6-s + (−1.99 − 2i)8-s + (−1.73 + i)9-s + (2.73 + 1.58i)10-s + (0.5 − 0.866i)11-s + (−1.58 + 1.58i)13-s + 5i·15-s + (−1.99 − 3.46i)16-s + (−2.15 + 0.578i)17-s + (−2.73 + 0.732i)18-s + (−1.58 − 2.73i)19-s + (1 − 0.999i)22-s + (0.732 − 2.73i)23-s + ⋯ |
L(s) = 1 | + (0.965 + 0.258i)2-s + (0.334 + 1.24i)3-s + (0.965 + 0.258i)5-s + 1.29i·6-s + (−0.707 − 0.707i)8-s + (−0.577 + 0.333i)9-s + (0.866 + 0.499i)10-s + (0.150 − 0.261i)11-s + (−0.438 + 0.438i)13-s + 1.29i·15-s + (−0.499 − 0.866i)16-s + (−0.523 + 0.140i)17-s + (−0.643 + 0.172i)18-s + (−0.362 − 0.628i)19-s + (0.213 − 0.213i)22-s + (0.152 − 0.569i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.347 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.347 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.79062 + 1.24571i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.79062 + 1.24571i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-2.15 - 0.578i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-1.36 - 0.366i)T + (1.73 + i)T^{2} \) |
| 3 | \( 1 + (-0.578 - 2.15i)T + (-2.59 + 1.5i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.58 - 1.58i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.15 - 0.578i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (1.58 + 2.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.732 + 2.73i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + 3iT - 29T^{2} \) |
| 31 | \( 1 + (2.73 + 1.58i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-8.19 - 2.19i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 9.48iT - 41T^{2} \) |
| 43 | \( 1 + (3 + 3i)T + 43iT^{2} \) |
| 47 | \( 1 + (1.73 - 6.47i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (1.36 - 0.366i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (4.74 - 8.21i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.47 + 3.16i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.366 + 1.36i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (11.2 - 6.5i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.16 - 3.16i)T - 83iT^{2} \) |
| 89 | \( 1 + (-3.16 - 5.47i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.58 + 1.58i)T + 97iT^{2} \) |
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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
−12.59581167495309949227173824172, −11.20386197139906617270038201865, −10.22017336040943852281917035571, −9.439056915875433132247461490440, −8.824817603429970577583569782402, −6.88021342916338866713574405906, −5.90019037673660997638812009207, −4.83782270874290395458197604686, −4.04951281563458854078929352157, −2.70866918238694926421179692792,
1.78579129101961203995678500212, 2.93447132185462453573220616223, 4.60884209275413838381972279032, 5.72375562106129228711743512995, 6.68898439328705372120485443322, 7.907590068083944077617168984594, 8.883627369187729293288007240860, 9.958686063473909863158491595016, 11.37794365596270044470644892081, 12.41657086325631448552315057606