| L(s) = 1 | + (0.809 − 0.587i)2-s + (0.913 − 0.406i)3-s + (0.309 − 0.951i)4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)6-s + (1.17 + 1.30i)7-s + (−0.309 − 0.951i)8-s + (0.669 − 0.743i)9-s + (0.913 + 0.406i)10-s + (2.60 − 0.552i)11-s + (−0.104 − 0.994i)12-s + (−0.429 + 4.08i)13-s + (1.71 + 0.365i)14-s + (0.809 + 0.587i)15-s + (−0.809 − 0.587i)16-s + (2.56 + 0.544i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (0.527 − 0.234i)3-s + (0.154 − 0.475i)4-s + (0.223 + 0.387i)5-s + (0.204 − 0.353i)6-s + (0.443 + 0.493i)7-s + (−0.109 − 0.336i)8-s + (0.223 − 0.247i)9-s + (0.288 + 0.128i)10-s + (0.784 − 0.166i)11-s + (−0.0301 − 0.287i)12-s + (−0.119 + 1.13i)13-s + (0.458 + 0.0975i)14-s + (0.208 + 0.151i)15-s + (−0.202 − 0.146i)16-s + (0.621 + 0.132i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.84549 - 0.734030i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.84549 - 0.734030i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.913 + 0.406i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (-3.41 + 4.39i)T \) |
| good | 7 | \( 1 + (-1.17 - 1.30i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (-2.60 + 0.552i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (0.429 - 4.08i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (-2.56 - 0.544i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (0.494 + 4.70i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-0.937 - 2.88i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (0.912 - 0.663i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (2.01 - 3.48i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.47 + 1.99i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (0.639 + 6.08i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (-1.96 - 1.42i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-2.35 + 2.61i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (12.1 - 5.40i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 - 2.77T + 61T^{2} \) |
| 67 | \( 1 + (-1.91 - 3.31i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.453 - 0.503i)T + (-7.42 - 70.6i)T^{2} \) |
| 73 | \( 1 + (3.03 - 0.645i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (0.829 + 0.176i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (7.09 + 3.16i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (-0.0570 + 0.175i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-2.90 + 8.93i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−9.945077425279620303904702826801, −9.208189142718202124391700271052, −8.512558054976500187630817008303, −7.26599590694044148062331953538, −6.59834673755094337701330597340, −5.60276851908254997721600991682, −4.53444206563482266542009821187, −3.55689050801291897422524632631, −2.48127967419263209448043987549, −1.51324535665563342219673586710,
1.41845626908878635339713547123, 2.95691412533920697969039695270, 3.93094796247649660953240825906, 4.80198731898136616419406400009, 5.68651124901706829153326381947, 6.69447443563320292033704791887, 7.76326510145632055152364381740, 8.225463060130417729759238811622, 9.223639222808262571458126564645, 10.12860073866604152298644220222
