| L(s) = 1 | + (1.40 + 0.122i)2-s + (0.988 − 0.149i)3-s + (1.96 + 0.346i)4-s + (−2.61 − 1.02i)5-s + (1.41 − 0.0885i)6-s + (2.63 − 0.209i)7-s + (2.73 + 0.729i)8-s + (0.955 − 0.294i)9-s + (−3.56 − 1.76i)10-s + (0.0741 − 0.240i)11-s + (1.99 + 0.0485i)12-s + (1.99 − 0.456i)13-s + (3.74 + 0.0288i)14-s + (−2.74 − 0.625i)15-s + (3.76 + 1.36i)16-s + (0.553 + 0.812i)17-s + ⋯ |
| L(s) = 1 | + (0.996 + 0.0868i)2-s + (0.570 − 0.0860i)3-s + (0.984 + 0.173i)4-s + (−1.17 − 0.459i)5-s + (0.576 − 0.0361i)6-s + (0.996 − 0.0791i)7-s + (0.966 + 0.257i)8-s + (0.318 − 0.0982i)9-s + (−1.12 − 0.559i)10-s + (0.0223 − 0.0724i)11-s + (0.577 + 0.0140i)12-s + (0.554 − 0.126i)13-s + (0.999 + 0.00770i)14-s + (−0.707 − 0.161i)15-s + (0.940 + 0.340i)16-s + (0.134 + 0.196i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.03935 - 0.300448i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.03935 - 0.300448i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.40 - 0.122i)T \) |
| 3 | \( 1 + (-0.988 + 0.149i)T \) |
| 7 | \( 1 + (-2.63 + 0.209i)T \) |
| good | 5 | \( 1 + (2.61 + 1.02i)T + (3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (-0.0741 + 0.240i)T + (-9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (-1.99 + 0.456i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-0.553 - 0.812i)T + (-6.21 + 15.8i)T^{2} \) |
| 19 | \( 1 + (2.11 + 3.65i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.14 - 6.07i)T + (-8.40 - 21.4i)T^{2} \) |
| 29 | \( 1 + (-0.989 - 0.476i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (-4.36 + 7.56i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.541 - 7.22i)T + (-36.5 + 5.51i)T^{2} \) |
| 41 | \( 1 + (4.29 - 3.42i)T + (9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (4.26 + 3.40i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (7.47 - 6.93i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (0.250 - 3.33i)T + (-52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (3.61 + 9.21i)T + (-43.2 + 40.1i)T^{2} \) |
| 61 | \( 1 + (8.77 - 0.657i)T + (60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (3.02 + 1.74i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.80 - 3.75i)T + (-44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (11.3 - 12.2i)T + (-5.45 - 72.7i)T^{2} \) |
| 79 | \( 1 + (7.19 - 4.15i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.432 + 1.89i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (0.340 + 1.10i)T + (-73.5 + 50.1i)T^{2} \) |
| 97 | \( 1 - 5.11iT - 97T^{2} \) |
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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
−11.15907237208068014229112023264, −9.885624773442806189425524768404, −8.363724177943063005525435399921, −8.088770755236064554437119790798, −7.20506292505443034682007893320, −6.00493660496253014882307690810, −4.73430816431153829179242079597, −4.15686664429609268549938554724, −3.10707653525933921145559693607, −1.57429670033395360774494584471,
1.82610959203929249646137729546, 3.16872168877066420300592734491, 4.06361798060425576745054394187, 4.79666688104888498199289184113, 6.15737237636432702585715664205, 7.17827534690478938114247355735, 7.995659163256313627906353516934, 8.603621522626506248465531004821, 10.30829430968271569541617625192, 10.78693969264152663446092547632
