| L(s) = 1 | + (−0.809 + 0.587i)2-s + (0.104 − 0.994i)3-s + (0.309 − 0.951i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s + (3.84 − 0.817i)7-s + (0.309 + 0.951i)8-s + (−0.978 − 0.207i)9-s + (0.104 + 0.994i)10-s + (2.81 + 3.12i)11-s + (−0.913 − 0.406i)12-s + (−1.77 + 0.788i)13-s + (−2.63 + 2.92i)14-s + (−0.809 − 0.587i)15-s + (−0.809 − 0.587i)16-s + (1.32 − 1.47i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (0.0603 − 0.574i)3-s + (0.154 − 0.475i)4-s + (0.223 − 0.387i)5-s + (0.204 + 0.353i)6-s + (1.45 − 0.308i)7-s + (0.109 + 0.336i)8-s + (−0.326 − 0.0693i)9-s + (0.0330 + 0.314i)10-s + (0.847 + 0.941i)11-s + (−0.263 − 0.117i)12-s + (−0.491 + 0.218i)13-s + (−0.702 + 0.780i)14-s + (−0.208 − 0.151i)15-s + (−0.202 − 0.146i)16-s + (0.321 − 0.357i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.121i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.57318 - 0.0962039i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.57318 - 0.0962039i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.809 - 0.587i)T \) |
| 3 | \( 1 + (-0.104 + 0.994i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + (2.17 + 5.12i)T \) |
| good | 7 | \( 1 + (-3.84 + 0.817i)T + (6.39 - 2.84i)T^{2} \) |
| 11 | \( 1 + (-2.81 - 3.12i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (1.77 - 0.788i)T + (8.69 - 9.66i)T^{2} \) |
| 17 | \( 1 + (-1.32 + 1.47i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-7.60 - 3.38i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (-2.36 - 7.27i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (5.80 - 4.22i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (4.54 + 7.86i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.0138 + 0.131i)T + (-40.1 + 8.52i)T^{2} \) |
| 43 | \( 1 + (-4.37 - 1.94i)T + (28.7 + 31.9i)T^{2} \) |
| 47 | \( 1 + (7.34 + 5.33i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-9.49 - 2.01i)T + (48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (-0.135 + 1.28i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 - 0.737T + 61T^{2} \) |
| 67 | \( 1 + (-7.52 + 13.0i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.75 + 0.797i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (-7.66 - 8.50i)T + (-7.63 + 72.6i)T^{2} \) |
| 79 | \( 1 + (6.64 - 7.38i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (0.579 + 5.51i)T + (-81.1 + 17.2i)T^{2} \) |
| 89 | \( 1 + (0.924 - 2.84i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-2.79 + 8.58i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−9.600663871889174614399775867145, −9.372945506600521006308573173992, −8.199753167052728840270956991994, −7.34681947507543347899326011831, −7.20921088655192454905483743921, −5.57554350513404709060233277098, −5.12981610813981308995692201544, −3.80193557100186373687455767637, −1.90792234538293037738185061383, −1.26795768209054232262551382526,
1.18789347732996117002490948551, 2.56397539821539925639476223166, 3.55792046556310969934540985904, 4.78360480928020620418816992221, 5.58062370903441408600806117580, 6.82787038656975109134804784949, 7.81442985738631370143275103313, 8.598024343647660915954697119276, 9.211414173833671251238644584838, 10.13779716922317047780923144490
