| L(s) = 1 | + (−1.30 − 0.835i)2-s + (0.142 + 0.989i)3-s + (0.162 + 0.355i)4-s + (0.959 + 0.281i)5-s + (0.642 − 1.40i)6-s + (1.57 − 1.81i)7-s + (−0.354 + 2.46i)8-s + (1.91 − 0.563i)9-s + (−1.01 − 1.16i)10-s + (3.19 − 2.05i)11-s + (−0.328 + 0.211i)12-s + (−1.07 − 1.23i)13-s + (−3.55 + 1.04i)14-s + (−0.142 + 0.989i)15-s + (3.03 − 3.49i)16-s + (−1.19 + 2.60i)17-s + ⋯ |
| L(s) = 1 | + (−0.919 − 0.591i)2-s + (0.0821 + 0.571i)3-s + (0.0811 + 0.177i)4-s + (0.429 + 0.125i)5-s + (0.262 − 0.574i)6-s + (0.593 − 0.684i)7-s + (−0.125 + 0.870i)8-s + (0.639 − 0.187i)9-s + (−0.320 − 0.369i)10-s + (0.963 − 0.619i)11-s + (−0.0948 + 0.0609i)12-s + (−0.296 − 0.342i)13-s + (−0.950 + 0.279i)14-s + (−0.0367 + 0.255i)15-s + (0.757 − 0.874i)16-s + (−0.288 + 0.632i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 + 0.594i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.804 + 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.736550 - 0.242754i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.736550 - 0.242754i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.959 - 0.281i)T \) |
| 23 | \( 1 + (3.50 - 3.27i)T \) |
| good | 2 | \( 1 + (1.30 + 0.835i)T + (0.830 + 1.81i)T^{2} \) |
| 3 | \( 1 + (-0.142 - 0.989i)T + (-2.87 + 0.845i)T^{2} \) |
| 7 | \( 1 + (-1.57 + 1.81i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-3.19 + 2.05i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (1.07 + 1.23i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.19 - 2.60i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (0.754 + 1.65i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (3.57 - 7.83i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.185 + 1.28i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (6.58 - 1.93i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (6.01 + 1.76i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (1.72 + 11.9i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 4.66T + 47T^{2} \) |
| 53 | \( 1 + (-3.35 + 3.86i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (4.75 + 5.48i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (1.23 - 8.57i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-4.36 - 2.80i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (2.38 + 1.53i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (0.872 + 1.90i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-8.09 - 9.34i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (13.1 - 3.84i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-1.65 - 11.5i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-5.95 - 1.74i)T + (81.6 + 52.4i)T^{2} \) |
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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
−13.61581580082426763266760005473, −12.09960963960549900371812002607, −10.90077102522599449095083239079, −10.36231616721328660712971500392, −9.371997401584068332655296130276, −8.503855053657814199497544800785, −7.02481889097418054872894896717, −5.33233567324296098300253193928, −3.78341123000989059494978050425, −1.56374161847528184207976961542,
1.85001603054626592117685022471, 4.43469900075623912426136651106, 6.26551571583723082422230471279, 7.23008356663628669120739544730, 8.239663147185042075623526392210, 9.254443132264763535425898100820, 10.07590803298307438123678700588, 11.80894335750320693566982193883, 12.56049229100915064158440580677, 13.69857327531907468357934183554
