L(s) = 1 | + (0.102 + 1.41i)2-s + (−1.97 + 0.288i)4-s + i·5-s + (−0.178 + 2.63i)7-s + (−0.608 − 2.76i)8-s + (−1.41 + 0.102i)10-s − 5.22i·11-s − 4.52i·13-s + (−3.74 + 0.0184i)14-s + (3.83 − 1.14i)16-s − 6.70i·17-s + 2.81·19-s + (−0.288 − 1.97i)20-s + (7.37 − 0.534i)22-s + 0.858i·23-s + ⋯ |
L(s) = 1 | + (0.0722 + 0.997i)2-s + (−0.989 + 0.144i)4-s + 0.447i·5-s + (−0.0673 + 0.997i)7-s + (−0.215 − 0.976i)8-s + (−0.446 + 0.0323i)10-s − 1.57i·11-s − 1.25i·13-s + (−0.999 + 0.00493i)14-s + (0.958 − 0.285i)16-s − 1.62i·17-s + 0.646·19-s + (−0.0644 − 0.442i)20-s + (1.57 − 0.113i)22-s + 0.179i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.208770525\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.208770525\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.102 - 1.41i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (0.178 - 2.63i)T \) |
good | 11 | \( 1 + 5.22iT - 11T^{2} \) |
| 13 | \( 1 + 4.52iT - 13T^{2} \) |
| 17 | \( 1 + 6.70iT - 17T^{2} \) |
| 19 | \( 1 - 2.81T + 19T^{2} \) |
| 23 | \( 1 - 0.858iT - 23T^{2} \) |
| 29 | \( 1 + 6.47T + 29T^{2} \) |
| 31 | \( 1 - 2.60T + 31T^{2} \) |
| 37 | \( 1 - 2.13T + 37T^{2} \) |
| 41 | \( 1 + 8.71iT - 41T^{2} \) |
| 43 | \( 1 - 7.42iT - 43T^{2} \) |
| 47 | \( 1 - 9.82T + 47T^{2} \) |
| 53 | \( 1 + 3.69T + 53T^{2} \) |
| 59 | \( 1 + 4.27T + 59T^{2} \) |
| 61 | \( 1 - 10.7iT - 61T^{2} \) |
| 67 | \( 1 + 4.52iT - 67T^{2} \) |
| 71 | \( 1 + 7.23iT - 71T^{2} \) |
| 73 | \( 1 + 9.24iT - 73T^{2} \) |
| 79 | \( 1 + 2.68iT - 79T^{2} \) |
| 83 | \( 1 - 16.2T + 83T^{2} \) |
| 89 | \( 1 + 8.53iT - 89T^{2} \) |
| 97 | \( 1 + 10.5iT - 97T^{2} \) |
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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
−9.325243034864486619317279468184, −8.902745615799658164250222827959, −7.88811027804460487363753290614, −7.38144180694784319412109424952, −6.14574267943048932951310756900, −5.69777783669948210637284963553, −4.96675166730519547669078668962, −3.44419704252452729428132305856, −2.83818603961341933917566398802, −0.54961278607871079833094207539,
1.32229293728490377202152071241, 2.15054605451249715314440952702, 3.78482833548462150793611484961, 4.24986922275899565068971523820, 5.07655472000800598595328459930, 6.34169193551903963308516793304, 7.35144155207902089400478789418, 8.157368509031191485255546799755, 9.219097735497375058629175644279, 9.734583275067334662594257515211