L(s) = 1 | + (0.587 − 0.809i)2-s + (0.809 + 0.587i)3-s + (−0.309 − 0.951i)4-s + (−0.309 − 0.951i)5-s + (0.951 − 0.309i)6-s + (−0.587 − 0.809i)7-s + (−0.951 − 0.309i)8-s + (0.309 + 0.951i)9-s + (−0.951 − 0.309i)10-s − i·11-s + (0.309 − 0.951i)12-s + 13-s − 14-s + (0.309 − 0.951i)15-s + (−0.809 + 0.587i)16-s + (0.951 − 0.309i)17-s + ⋯ |
L(s) = 1 | + (0.587 − 0.809i)2-s + (0.809 + 0.587i)3-s + (−0.309 − 0.951i)4-s + (−0.309 − 0.951i)5-s + (0.951 − 0.309i)6-s + (−0.587 − 0.809i)7-s + (−0.951 − 0.309i)8-s + (0.309 + 0.951i)9-s + (−0.951 − 0.309i)10-s − i·11-s + (0.309 − 0.951i)12-s + 13-s − 14-s + (0.309 − 0.951i)15-s + (−0.809 + 0.587i)16-s + (0.951 − 0.309i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.338183594 - 1.924887076i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.338183594 - 1.924887076i\) |
\(L(1)\) |
\(\approx\) |
\(1.353972449 - 0.9322737156i\) |
\(L(1)\) |
\(\approx\) |
\(1.353972449 - 0.9322737156i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 61 | \( 1 \) |
good | 2 | \( 1 + (0.587 - 0.809i)T \) |
| 3 | \( 1 + (0.809 + 0.587i)T \) |
| 5 | \( 1 + (-0.309 - 0.951i)T \) |
| 7 | \( 1 + (-0.587 - 0.809i)T \) |
| 11 | \( 1 - iT \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + (0.951 - 0.309i)T \) |
| 19 | \( 1 + (0.809 + 0.587i)T \) |
| 23 | \( 1 + (-0.951 + 0.309i)T \) |
| 29 | \( 1 - iT \) |
| 31 | \( 1 + (0.587 + 0.809i)T \) |
| 37 | \( 1 + (0.587 + 0.809i)T \) |
| 41 | \( 1 + (0.809 - 0.587i)T \) |
| 43 | \( 1 + (-0.951 - 0.309i)T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + (0.951 + 0.309i)T \) |
| 59 | \( 1 + (-0.587 + 0.809i)T \) |
| 67 | \( 1 + (0.951 - 0.309i)T \) |
| 71 | \( 1 + (-0.951 - 0.309i)T \) |
| 73 | \( 1 + (0.309 - 0.951i)T \) |
| 79 | \( 1 + (0.951 + 0.309i)T \) |
| 83 | \( 1 + (-0.809 - 0.587i)T \) |
| 89 | \( 1 + (-0.587 + 0.809i)T \) |
| 97 | \( 1 + (0.809 - 0.587i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−32.47126325448455980998521585598, −31.4097498566201178017663646230, −30.66526872582716308605679692550, −29.895425734274843671372279773, −28.05481000217349166243659500932, −26.35067294821649362430986185860, −25.80612112096674007449720352569, −24.96986964457012986495355896364, −23.59861839977087063452379877422, −22.73019190396107588314580974669, −21.541907166228847992774205576856, −20.09789621234350956554879834612, −18.61088335734234175243274093793, −17.9792660434558167366135078288, −15.95806949922015016943619681199, −15.075312976433463780250671454273, −14.164621788944932131941029624597, −12.90268856313529726630270791005, −11.8641579704300607204781152766, −9.59194639393865433214939530567, −8.16535252055528924560771631405, −7.06921896376327045618910039321, −5.99899207748416571634677703853, −3.77606091870044792197265947554, −2.62889655673528986470610526529,
1.06191087791363180198707646871, 3.29207599617430988123054809318, 4.116935753128568424795582140691, 5.685779473610725071092225427798, 8.08536798523911227733911957825, 9.38992967784235095071040248260, 10.445793629680005730858117493577, 11.91776680433701964768911387539, 13.45744133410103146383389426777, 13.92254798497392190170475375356, 15.69419175260461754299768885065, 16.4996637373539664816808411141, 18.76724706566584320563160806526, 19.78114724218851045606291450183, 20.58673527498668563134077735475, 21.378980087627349804346920733, 22.80448132521864208574031665804, 23.88284867816259311320979117877, 25.10877558043454035738837708518, 26.629191083932337079591385880407, 27.568118631258420452268737700629, 28.647828795082899622916955899786, 29.85505469732386909764234079170, 30.95014936617194186846182738692, 32.24707774734274823726808381430