| L(s) = 1 | + (−1.24 + 1.68i)2-s + (−2.33 + 0.441i)3-s + (−0.697 − 2.26i)4-s + (−2.23 − 0.156i)5-s + (2.15 − 4.46i)6-s + (−1.67 + 2.04i)7-s + (0.724 + 0.253i)8-s + (2.44 − 0.961i)9-s + (3.03 − 3.55i)10-s + (0.785 − 2.00i)11-s + (2.62 + 4.96i)12-s + (4.20 + 0.473i)13-s + (−1.35 − 5.36i)14-s + (5.26 − 0.619i)15-s + (2.58 − 1.76i)16-s + (−1.85 − 0.0695i)17-s + ⋯ |
| L(s) = 1 | + (−0.877 + 1.18i)2-s + (−1.34 + 0.254i)3-s + (−0.348 − 1.13i)4-s + (−0.997 − 0.0700i)5-s + (0.878 − 1.82i)6-s + (−0.634 + 0.773i)7-s + (0.256 + 0.0896i)8-s + (0.816 − 0.320i)9-s + (0.958 − 1.12i)10-s + (0.236 − 0.603i)11-s + (0.757 + 1.43i)12-s + (1.16 + 0.131i)13-s + (−0.362 − 1.43i)14-s + (1.36 − 0.159i)15-s + (0.646 − 0.440i)16-s + (−0.450 − 0.0168i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.167440 - 0.0124973i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.167440 - 0.0124973i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.23 + 0.156i)T \) |
| 7 | \( 1 + (1.67 - 2.04i)T \) |
| good | 2 | \( 1 + (1.24 - 1.68i)T + (-0.589 - 1.91i)T^{2} \) |
| 3 | \( 1 + (2.33 - 0.441i)T + (2.79 - 1.09i)T^{2} \) |
| 11 | \( 1 + (-0.785 + 2.00i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (-4.20 - 0.473i)T + (12.6 + 2.89i)T^{2} \) |
| 17 | \( 1 + (1.85 + 0.0695i)T + (16.9 + 1.27i)T^{2} \) |
| 19 | \( 1 + (3.30 - 5.72i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.262 + 7.01i)T + (-22.9 + 1.71i)T^{2} \) |
| 29 | \( 1 + (-4.54 + 1.03i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (3.46 - 1.99i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.342 - 0.180i)T + (20.8 - 30.5i)T^{2} \) |
| 41 | \( 1 + (4.04 + 8.40i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (3.00 + 8.58i)T + (-33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (4.09 + 3.02i)T + (13.8 + 44.9i)T^{2} \) |
| 53 | \( 1 + (-0.148 - 0.0786i)T + (29.8 + 43.7i)T^{2} \) |
| 59 | \( 1 + (0.00101 + 0.0136i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (-3.58 + 11.6i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-4.88 - 1.30i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (1.72 - 7.57i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-5.89 + 4.35i)T + (21.5 - 69.7i)T^{2} \) |
| 79 | \( 1 + (5.78 + 3.34i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.34 - 11.9i)T + (-80.9 + 18.4i)T^{2} \) |
| 89 | \( 1 + (0.169 + 0.432i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (7.26 + 7.26i)T + 97iT^{2} \) |
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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
−12.01601403354931607980252340861, −10.99012188085329801004095372779, −10.17535894864725728885321347251, −8.633267320177946730715587689513, −8.444086676529992751830558740710, −6.74092184217255592045190334345, −6.28841172517983115895199281692, −5.31985683804018998724881762217, −3.74890257929561437057362229798, −0.26084742556146350141059789766,
1.10117821158153632838409199031, 3.29318054435875218983654955378, 4.52390321764828513882375428011, 6.24761809132415478005788495766, 7.12967709370191621825938582707, 8.406216326411728750112268958091, 9.537610439515665619673990661769, 10.58040981864368628479599123106, 11.24187153653947568658615899360, 11.60286666625971340940824397861
