L(s) = 1 | + (−0.417 + 1.35i)2-s + (−1.00 + 0.580i)3-s + (−1.65 − 1.12i)4-s + 5-s + (−0.364 − 1.60i)6-s + (−2.74 − 1.58i)7-s + (2.21 − 1.76i)8-s + (−0.826 + 1.43i)9-s + (−0.417 + 1.35i)10-s + (−1.23 − 2.13i)11-s + (2.31 + 0.175i)12-s + (3.60 − 0.150i)13-s + (3.28 − 3.04i)14-s + (−1.00 + 0.580i)15-s + (1.45 + 3.72i)16-s + (0.369 − 0.639i)17-s + ⋯ |
L(s) = 1 | + (−0.295 + 0.955i)2-s + (−0.580 + 0.335i)3-s + (−0.825 − 0.564i)4-s + 0.447·5-s + (−0.148 − 0.653i)6-s + (−1.03 − 0.598i)7-s + (0.782 − 0.622i)8-s + (−0.275 + 0.477i)9-s + (−0.132 + 0.427i)10-s + (−0.371 − 0.643i)11-s + (0.668 + 0.0506i)12-s + (0.999 − 0.0416i)13-s + (0.877 − 0.813i)14-s + (−0.259 + 0.149i)15-s + (0.363 + 0.931i)16-s + (0.0895 − 0.155i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.176i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.789850 + 0.0700978i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.789850 + 0.0700978i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.417 - 1.35i)T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + (-3.60 + 0.150i)T \) |
good | 3 | \( 1 + (1.00 - 0.580i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (2.74 + 1.58i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.23 + 2.13i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.369 + 0.639i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.31 + 7.47i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.88 - 4.99i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.59 + 0.919i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 9.05iT - 31T^{2} \) |
| 37 | \( 1 + (-1.35 - 2.35i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.165 + 0.0952i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-7.43 - 4.29i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 8.23iT - 47T^{2} \) |
| 53 | \( 1 + 4.18iT - 53T^{2} \) |
| 59 | \( 1 + (-5.72 + 9.91i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.03 + 2.32i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.66 + 2.88i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.24 + 1.87i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 1.03iT - 73T^{2} \) |
| 79 | \( 1 + 9.18T + 79T^{2} \) |
| 83 | \( 1 - 1.79T + 83T^{2} \) |
| 89 | \( 1 + (-12.1 + 7.04i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (14.8 + 8.56i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.86436116292131413821118348168, −9.763780151243386260499590309574, −9.294603444756853887049708910927, −8.117923761009004618101914412745, −7.16777087203406723294038441384, −6.18277077057870354647374849099, −5.57855847370836146455397900055, −4.54511139913823067298989013877, −3.13805770120695634449426061513, −0.65891489725841402378564813176,
1.23923588230200569902590736599, 2.77182955663918343285431579113, 3.75501491186510882235311734467, 5.33789166170732443574379479574, 6.10278547030343444839680252753, 7.18072753894081425954899925087, 8.554145540660360025307055018713, 9.186102311886725946143713394255, 10.14719640343093552784391773900, 10.70408593149410961255691857440