| L(s) = 1 | + (−0.154 + 0.178i)2-s + (−0.841 + 0.540i)3-s + (0.276 + 1.92i)4-s + (0.418 + 0.916i)5-s + (0.0336 − 0.234i)6-s + (1.97 − 0.579i)7-s + (−0.784 − 0.504i)8-s + (0.415 − 0.909i)9-s + (−0.228 − 0.0670i)10-s + (−2.87 − 3.31i)11-s + (−1.27 − 1.46i)12-s + (2.00 + 0.588i)13-s + (−0.201 + 0.442i)14-s + (−0.847 − 0.544i)15-s + (−3.51 + 1.03i)16-s + (0.308 − 2.14i)17-s + ⋯ |
| L(s) = 1 | + (−0.109 + 0.126i)2-s + (−0.485 + 0.312i)3-s + (0.138 + 0.962i)4-s + (0.187 + 0.409i)5-s + (0.0137 − 0.0955i)6-s + (0.745 − 0.218i)7-s + (−0.277 − 0.178i)8-s + (0.138 − 0.303i)9-s + (−0.0722 − 0.0212i)10-s + (−0.866 − 1.00i)11-s + (−0.367 − 0.424i)12-s + (0.555 + 0.163i)13-s + (−0.0539 + 0.118i)14-s + (−0.218 − 0.140i)15-s + (−0.879 + 0.258i)16-s + (0.0748 − 0.520i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.724593 + 0.383979i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.724593 + 0.383979i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.841 - 0.540i)T \) |
| 23 | \( 1 + (-4.79 - 0.0634i)T \) |
| good | 2 | \( 1 + (0.154 - 0.178i)T + (-0.284 - 1.97i)T^{2} \) |
| 5 | \( 1 + (-0.418 - 0.916i)T + (-3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (-1.97 + 0.579i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (2.87 + 3.31i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.00 - 0.588i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.308 + 2.14i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (0.379 + 2.64i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (1.13 - 7.89i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (6.58 + 4.23i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (1.63 - 3.57i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-2.62 - 5.73i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (1.82 - 1.17i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + 7.22T + 47T^{2} \) |
| 53 | \( 1 + (-10.8 + 3.18i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (11.6 + 3.43i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (11.8 + 7.63i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (0.466 - 0.538i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-2.69 + 3.11i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.00834 - 0.0580i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-15.6 - 4.59i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (6.04 - 13.2i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-2.31 + 1.48i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-2.45 - 5.37i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−15.06956952741011488419105779645, −13.75277410069575824770256985548, −12.74419990112321664243383845939, −11.27470003612384540817205979091, −10.86970470774264098055702158022, −9.055469735066911416216212528256, −7.899988465376107319944111509443, −6.62811396623855534629556535591, −4.97372659023992932617676020665, −3.17786159089329656973945215190,
1.75989173279955733901018825534, 4.90049856169907381687740101634, 5.83762155503883520006997099509, 7.42881441692587607430871451065, 8.924017353224379718722666742063, 10.29822121215732263057672896183, 11.09256928106993691602479491964, 12.36147951999719230438456503639, 13.39683237039072369051982184759, 14.74765170996657570381913134080
