L(s) = 1 | + (−0.168 + 1.40i)2-s + (0.0286 + 0.190i)3-s + (−1.94 − 0.473i)4-s + (3.07 + 1.20i)5-s + (−0.271 + 0.00814i)6-s + (0.384 − 2.61i)7-s + (0.993 − 2.64i)8-s + (2.83 − 0.873i)9-s + (−2.21 + 4.10i)10-s + (0.799 − 2.59i)11-s + (0.0344 − 0.382i)12-s + (2.39 − 0.546i)13-s + (3.61 + 0.981i)14-s + (−0.141 + 0.618i)15-s + (3.55 + 1.84i)16-s + (−4.97 + 3.39i)17-s + ⋯ |
L(s) = 1 | + (−0.119 + 0.992i)2-s + (0.0165 + 0.109i)3-s + (−0.971 − 0.236i)4-s + (1.37 + 0.538i)5-s + (−0.110 + 0.00332i)6-s + (0.145 − 0.989i)7-s + (0.351 − 0.936i)8-s + (0.943 − 0.291i)9-s + (−0.698 + 1.29i)10-s + (0.241 − 0.781i)11-s + (0.00993 − 0.110i)12-s + (0.663 − 0.151i)13-s + (0.965 + 0.262i)14-s + (−0.0364 + 0.159i)15-s + (0.887 + 0.460i)16-s + (−1.20 + 0.823i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.570 - 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.570 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.39251 + 0.727891i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.39251 + 0.727891i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.168 - 1.40i)T \) |
| 7 | \( 1 + (-0.384 + 2.61i)T \) |
good | 3 | \( 1 + (-0.0286 - 0.190i)T + (-2.86 + 0.884i)T^{2} \) |
| 5 | \( 1 + (-3.07 - 1.20i)T + (3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (-0.799 + 2.59i)T + (-9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (-2.39 + 0.546i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (4.97 - 3.39i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (-2.12 + 1.22i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.24 + 2.89i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (2.12 - 4.40i)T + (-18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (2.86 - 4.95i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.51 + 0.263i)T + (36.5 - 5.51i)T^{2} \) |
| 41 | \( 1 + (1.26 + 1.58i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (-8.29 - 6.61i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (4.91 - 4.56i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (11.7 + 0.883i)T + (52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (-8.66 + 3.40i)T + (43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (2.16 - 0.162i)T + (60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (4.07 + 2.35i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (7.32 - 3.52i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-1.71 - 1.58i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (4.47 + 7.75i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.55 + 0.812i)T + (74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (-14.8 + 4.56i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 - 8.27T + 97T^{2} \) |
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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
−10.95709886943878726571144865514, −10.41117568494321161105624057500, −9.555706051203576054477787708932, −8.731005003180914569252471180560, −7.52979588426074883857094193280, −6.50746337312294625519927231715, −6.11272753490560078090531958365, −4.70269699466270873716151687629, −3.60812600151356245627701334182, −1.42646286105449883095672837647,
1.67046325936504229062797184433, 2.30348957709242601136025604482, 4.18641272327954542845442568114, 5.17517788312196498520224599153, 6.13161208659588492494370165552, 7.64265592832226415942245688727, 8.889852843732907188209033493360, 9.500362163391069085001420473057, 10.00922140816049679054847284606, 11.25303259358708124056496841066