| L(s) = 1 | + (−0.795 − 1.16i)2-s + (0.172 − 2.40i)3-s + (−0.734 + 1.86i)4-s + (2.05 − 0.881i)5-s + (−2.95 + 1.71i)6-s + (−1.70 − 4.57i)7-s + (2.75 − 0.619i)8-s + (−2.80 − 0.403i)9-s + (−2.66 − 1.70i)10-s + (−0.203 + 0.693i)11-s + (4.35 + 2.09i)12-s + (4.27 + 1.59i)13-s + (−3.99 + 5.63i)14-s + (−1.76 − 5.10i)15-s + (−2.91 − 2.73i)16-s + (−0.642 − 2.95i)17-s + ⋯ |
| L(s) = 1 | + (−0.562 − 0.826i)2-s + (0.0994 − 1.39i)3-s + (−0.367 + 0.930i)4-s + (0.919 − 0.394i)5-s + (−1.20 + 0.699i)6-s + (−0.645 − 1.73i)7-s + (0.975 − 0.219i)8-s + (−0.934 − 0.134i)9-s + (−0.842 − 0.538i)10-s + (−0.0613 + 0.209i)11-s + (1.25 + 0.603i)12-s + (1.18 + 0.442i)13-s + (−1.06 + 1.50i)14-s + (−0.456 − 1.31i)15-s + (−0.729 − 0.683i)16-s + (−0.155 − 0.715i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.128i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0728603 + 1.13058i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0728603 + 1.13058i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.795 + 1.16i)T \) |
| 5 | \( 1 + (-2.05 + 0.881i)T \) |
| 23 | \( 1 + (3.04 - 3.70i)T \) |
| good | 3 | \( 1 + (-0.172 + 2.40i)T + (-2.96 - 0.426i)T^{2} \) |
| 7 | \( 1 + (1.70 + 4.57i)T + (-5.29 + 4.58i)T^{2} \) |
| 11 | \( 1 + (0.203 - 0.693i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-4.27 - 1.59i)T + (9.82 + 8.51i)T^{2} \) |
| 17 | \( 1 + (0.642 + 2.95i)T + (-15.4 + 7.06i)T^{2} \) |
| 19 | \( 1 + (0.396 - 0.254i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-0.693 + 1.07i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-4.36 - 3.77i)T + (4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (7.79 - 5.83i)T + (10.4 - 35.5i)T^{2} \) |
| 41 | \( 1 + (-0.653 - 4.54i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (0.604 - 8.45i)T + (-42.5 - 6.11i)T^{2} \) |
| 47 | \( 1 + (-6.70 + 6.70i)T - 47iT^{2} \) |
| 53 | \( 1 + (-0.0196 - 0.0526i)T + (-40.0 + 34.7i)T^{2} \) |
| 59 | \( 1 + (-2.37 - 5.20i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (0.0965 - 0.111i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (0.00376 - 0.00689i)T + (-36.2 - 56.3i)T^{2} \) |
| 71 | \( 1 + (-0.0265 - 0.0903i)T + (-59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-3.04 + 13.9i)T + (-66.4 - 30.3i)T^{2} \) |
| 79 | \( 1 + (0.0331 + 0.0726i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-7.17 + 5.36i)T + (23.3 - 79.6i)T^{2} \) |
| 89 | \( 1 + (-10.3 + 8.97i)T + (12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (2.65 - 3.55i)T + (-27.3 - 93.0i)T^{2} \) |
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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
−10.44689684470667134811029058860, −9.861179459004449689749753773412, −8.820935601541598251715864431515, −7.86263739452756540497825837046, −6.99483921541498946611101562113, −6.32883390165385896391099856093, −4.50833873365920651479088872989, −3.22972647117375299939407927148, −1.73529622350207857659495787898, −0.897150167487653844341552914865,
2.32667428451159855403011658550, 3.76885326301889126980095307458, 5.30106239660295919978055426721, 5.83852211900195070360845265075, 6.56679913949061148628528996746, 8.394933077827494287958209679335, 8.911739831437884147359672666459, 9.563402001373542991625865029383, 10.41146176767936119456683038447, 10.90511606661780267828524057761
