| L(s) = 1 | + (0.369 − 1.13i)2-s + (−2.36 + 0.934i)3-s + (0.462 + 0.335i)4-s + (−1.04 + 0.981i)5-s + (0.190 + 3.02i)6-s + (3.65 + 2.00i)7-s + (2.48 − 1.80i)8-s + (2.51 − 2.36i)9-s + (0.729 + 1.54i)10-s + (1.80 + 0.712i)11-s + (−1.40 − 0.360i)12-s + (−3.50 + 5.52i)13-s + (3.63 − 3.41i)14-s + (1.54 − 3.29i)15-s + (−0.781 − 2.40i)16-s + (2.82 − 1.55i)17-s + ⋯ |
| L(s) = 1 | + (0.261 − 0.803i)2-s + (−1.36 + 0.539i)3-s + (0.231 + 0.167i)4-s + (−0.467 + 0.438i)5-s + (0.0778 + 1.23i)6-s + (1.38 + 0.759i)7-s + (0.879 − 0.638i)8-s + (0.838 − 0.787i)9-s + (0.230 + 0.490i)10-s + (0.542 + 0.214i)11-s + (−0.405 − 0.104i)12-s + (−0.973 + 1.53i)13-s + (0.970 − 0.911i)14-s + (0.400 − 0.850i)15-s + (−0.195 − 0.601i)16-s + (0.684 − 0.376i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.341i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.979334 + 0.172557i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.979334 + 0.172557i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 + (-4.77 - 11.3i)T \) |
| good | 2 | \( 1 + (-0.369 + 1.13i)T + (-1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (2.36 - 0.934i)T + (2.18 - 2.05i)T^{2} \) |
| 5 | \( 1 + (1.04 - 0.981i)T + (0.313 - 4.99i)T^{2} \) |
| 7 | \( 1 + (-3.65 - 2.00i)T + (3.75 + 5.91i)T^{2} \) |
| 11 | \( 1 + (-1.80 - 0.712i)T + (8.01 + 7.53i)T^{2} \) |
| 13 | \( 1 + (3.50 - 5.52i)T + (-5.53 - 11.7i)T^{2} \) |
| 17 | \( 1 + (-2.82 + 1.55i)T + (9.10 - 14.3i)T^{2} \) |
| 19 | \( 1 + (4.30 + 3.13i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (1.04 + 0.762i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.87 + 3.47i)T + (-5.43 - 28.4i)T^{2} \) |
| 31 | \( 1 + (0.755 + 3.96i)T + (-28.8 + 11.4i)T^{2} \) |
| 37 | \( 1 + (2.08 - 3.29i)T + (-15.7 - 33.4i)T^{2} \) |
| 41 | \( 1 + (0.294 + 4.68i)T + (-40.6 + 5.13i)T^{2} \) |
| 43 | \( 1 + (-9.95 + 5.47i)T + (23.0 - 36.3i)T^{2} \) |
| 47 | \( 1 + (0.246 + 3.92i)T + (-46.6 + 5.89i)T^{2} \) |
| 53 | \( 1 + (11.4 - 2.94i)T + (46.4 - 25.5i)T^{2} \) |
| 59 | \( 1 + (1.70 + 5.25i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (4.74 + 1.87i)T + (44.4 + 41.7i)T^{2} \) |
| 67 | \( 1 + (-3.50 - 3.28i)T + (4.20 + 66.8i)T^{2} \) |
| 71 | \( 1 + (10.8 + 5.94i)T + (38.0 + 59.9i)T^{2} \) |
| 73 | \( 1 + (-10.5 - 5.79i)T + (39.1 + 61.6i)T^{2} \) |
| 79 | \( 1 + (-2.14 + 11.2i)T + (-73.4 - 29.0i)T^{2} \) |
| 83 | \( 1 + (-7.64 - 0.965i)T + (80.3 + 20.6i)T^{2} \) |
| 89 | \( 1 + (-11.8 - 1.49i)T + (86.2 + 22.1i)T^{2} \) |
| 97 | \( 1 + (4.07 + 3.82i)T + (6.09 + 96.8i)T^{2} \) |
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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
−12.28923532301324478995336329091, −11.82356730801531289188527944618, −11.33987745576830977585777677946, −10.58411478872074892551044476119, −9.294792873036430564301466219046, −7.61869225059557898962795976961, −6.51335320316988667538146514645, −4.94950312350356745396130439498, −4.20260514699144306871822777849, −2.12574511296553913715349124940,
1.24175926343646575568998486140, 4.55147430424918522832190917630, 5.37438400873798459764824488879, 6.36777673077026766227009500282, 7.61226942506486343797586902180, 8.077232196312487153894610850752, 10.47143013745900238832703161213, 10.92410749832006599945498222945, 12.04356937857984078567489531992, 12.68632489504465195707809358376
