| L(s) = 1 | + (−1.75 − 1.27i)2-s + (−0.104 − 0.994i)3-s + (0.831 + 2.55i)4-s + (−1.24 − 2.15i)5-s + (−1.08 + 1.87i)6-s + (−1.22 − 0.260i)7-s + (0.462 − 1.42i)8-s + (−0.978 + 0.207i)9-s + (−0.563 + 5.36i)10-s + (−0.461 + 0.512i)11-s + (2.45 − 1.09i)12-s + (−4.28 − 1.90i)13-s + (1.81 + 2.01i)14-s + (−2.01 + 1.46i)15-s + (1.73 − 1.25i)16-s + (0.123 + 0.137i)17-s + ⋯ |
| L(s) = 1 | + (−1.23 − 0.900i)2-s + (−0.0603 − 0.574i)3-s + (0.415 + 1.27i)4-s + (−0.556 − 0.964i)5-s + (−0.442 + 0.765i)6-s + (−0.462 − 0.0983i)7-s + (0.163 − 0.503i)8-s + (−0.326 + 0.0693i)9-s + (−0.178 + 1.69i)10-s + (−0.139 + 0.154i)11-s + (0.709 − 0.316i)12-s + (−1.18 − 0.528i)13-s + (0.484 + 0.538i)14-s + (−0.520 + 0.378i)15-s + (0.432 − 0.314i)16-s + (0.0300 + 0.0333i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00896539 - 0.379936i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00896539 - 0.379936i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.104 + 0.994i)T \) |
| 31 | \( 1 + (5.18 + 2.01i)T \) |
| good | 2 | \( 1 + (1.75 + 1.27i)T + (0.618 + 1.90i)T^{2} \) |
| 5 | \( 1 + (1.24 + 2.15i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (1.22 + 0.260i)T + (6.39 + 2.84i)T^{2} \) |
| 11 | \( 1 + (0.461 - 0.512i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (4.28 + 1.90i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (-0.123 - 0.137i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-7.04 + 3.13i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-2.57 + 7.94i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-3.57 - 2.59i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (0.974 - 1.68i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.893 - 8.50i)T + (-40.1 - 8.52i)T^{2} \) |
| 43 | \( 1 + (-9.84 + 4.38i)T + (28.7 - 31.9i)T^{2} \) |
| 47 | \( 1 + (3.76 - 2.73i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-8.21 + 1.74i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (0.604 + 5.75i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + 3.09T + 61T^{2} \) |
| 67 | \( 1 + (-5.04 - 8.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-9.19 + 1.95i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (7.51 - 8.34i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (0.141 + 0.156i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (-0.756 + 7.19i)T + (-81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (0.741 + 2.28i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (1.13 + 3.50i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−12.88554093916057920128323315384, −12.33956592671372788599935165212, −11.40415008668974507560259593791, −10.14573785751072473131964476650, −9.165914358483285365566315469981, −8.172637862789206913134079740528, −7.17701965654419065249164743203, −5.04230677409275447925266824108, −2.79074838526034216579059416186, −0.68911771498467960122065553529,
3.40209829454563195166305817061, 5.57910930521321364249281392374, 7.06355540119700930586785916146, 7.66081807739378419158286938652, 9.238360258945841031521130527510, 9.866323046339224372150043916105, 10.98256406861811562973012097521, 12.14112493930669300748336029175, 14.02899964117801355036850743067, 14.99255193360981836215942513812
