| L(s) = 1 | + (−0.376 + 1.36i)2-s + (−0.971 − 1.43i)3-s + (−1.71 − 1.02i)4-s + (1.29 − 3.56i)5-s + (2.32 − 0.785i)6-s + (−2.58 − 0.455i)7-s + (2.04 − 1.95i)8-s + (−1.11 + 2.78i)9-s + (4.37 + 3.10i)10-s + (3.39 − 1.23i)11-s + (0.197 + 3.45i)12-s + (−0.819 + 0.688i)13-s + (1.59 − 3.34i)14-s + (−6.37 + 1.60i)15-s + (1.89 + 3.52i)16-s + (0.980 − 0.566i)17-s + ⋯ |
| L(s) = 1 | + (−0.265 + 0.963i)2-s + (−0.561 − 0.827i)3-s + (−0.858 − 0.512i)4-s + (0.580 − 1.59i)5-s + (0.947 − 0.320i)6-s + (−0.975 − 0.171i)7-s + (0.722 − 0.691i)8-s + (−0.370 + 0.928i)9-s + (1.38 + 0.983i)10-s + (1.02 − 0.372i)11-s + (0.0571 + 0.998i)12-s + (−0.227 + 0.190i)13-s + (0.425 − 0.894i)14-s + (−1.64 + 0.413i)15-s + (0.474 + 0.880i)16-s + (0.237 − 0.137i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.618 + 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.618 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.628896 - 0.305412i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.628896 - 0.305412i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.376 - 1.36i)T \) |
| 3 | \( 1 + (0.971 + 1.43i)T \) |
| good | 5 | \( 1 + (-1.29 + 3.56i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (2.58 + 0.455i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-3.39 + 1.23i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.819 - 0.688i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.980 + 0.566i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.0627 - 0.0362i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.0731 + 0.414i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.192 - 0.229i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-6.88 + 1.21i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.377 - 0.654i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.36 - 7.58i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.18 - 8.74i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.0443 + 0.251i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 12.0iT - 53T^{2} \) |
| 59 | \( 1 + (-11.7 - 4.27i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (1.13 - 6.43i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (4.37 + 5.21i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (6.35 + 11.0i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.578 - 1.00i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.09 - 6.07i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (3.00 + 2.51i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-2.07 - 1.19i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.81 + 0.660i)T + (74.3 - 62.3i)T^{2} \) |
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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
−13.36515815089716874363647290919, −12.89422981217243684995226604861, −11.79288159219140895660684599937, −9.934516120938727922325101701900, −9.091700372748980124290168605698, −8.045117663477607556953766916992, −6.60600868428790983038890120382, −5.84799399104547489705624208891, −4.59259136333057025605146342998, −1.02302464959463052039374838816,
2.80774029741581030416450137897, 3.94325040765763043681910840541, 5.86088605003234183011507836347, 7.00401104830683475527261789605, 9.117643664670104044045184940988, 9.948457727999988631218643423203, 10.51164974248554213567580058583, 11.55213787757870123853972997070, 12.47637664502922651967784449486, 13.91722339056900659354796611613
