| L(s) = 1 | + (0.258 − 2.45i)2-s + (−1.04 − 0.221i)3-s + (−4.02 − 0.854i)4-s + (1.10 − 1.94i)5-s + (−0.813 + 2.50i)6-s + (−2.33 + 1.25i)7-s + (−1.61 + 4.96i)8-s + (−1.70 − 0.758i)9-s + (−4.49 − 3.21i)10-s + (4.96 − 2.20i)11-s + (4.00 + 1.78i)12-s + (−2.83 + 2.05i)13-s + (2.47 + 6.05i)14-s + (−1.58 + 1.78i)15-s + (4.27 + 1.90i)16-s + (0.0152 + 0.0168i)17-s + ⋯ |
| L(s) = 1 | + (0.182 − 1.73i)2-s + (−0.601 − 0.127i)3-s + (−2.01 − 0.427i)4-s + (0.493 − 0.869i)5-s + (−0.332 + 1.02i)6-s + (−0.881 + 0.472i)7-s + (−0.570 + 1.75i)8-s + (−0.567 − 0.252i)9-s + (−1.42 − 1.01i)10-s + (1.49 − 0.666i)11-s + (1.15 + 0.514i)12-s + (−0.784 + 0.570i)13-s + (0.660 + 1.61i)14-s + (−0.408 + 0.460i)15-s + (1.06 + 0.475i)16-s + (0.00368 + 0.00409i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 - 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.915 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.173262 + 0.822682i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.173262 + 0.822682i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.10 + 1.94i)T \) |
| 7 | \( 1 + (2.33 - 1.25i)T \) |
| good | 2 | \( 1 + (-0.258 + 2.45i)T + (-1.95 - 0.415i)T^{2} \) |
| 3 | \( 1 + (1.04 + 0.221i)T + (2.74 + 1.22i)T^{2} \) |
| 11 | \( 1 + (-4.96 + 2.20i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (2.83 - 2.05i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.0152 - 0.0168i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-5.11 + 1.08i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (-0.721 + 6.85i)T + (-22.4 - 4.78i)T^{2} \) |
| 29 | \( 1 + (1.64 + 5.06i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.42 - 1.58i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (2.61 + 1.16i)T + (24.7 + 27.4i)T^{2} \) |
| 41 | \( 1 + (-1.11 + 0.807i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 6.93T + 43T^{2} \) |
| 47 | \( 1 + (-0.0450 + 0.0500i)T + (-4.91 - 46.7i)T^{2} \) |
| 53 | \( 1 + (-1.86 - 0.395i)T + (48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (-1.58 - 15.0i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (0.0491 - 0.467i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (-4.24 - 4.71i)T + (-7.00 + 66.6i)T^{2} \) |
| 71 | \( 1 + (-1.37 - 4.21i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-1.22 + 0.545i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-0.558 + 0.620i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-3.38 + 10.4i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (0.781 - 7.43i)T + (-87.0 - 18.5i)T^{2} \) |
| 97 | \( 1 + (-4.86 - 14.9i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.96303217912733675450121987579, −11.62923742819943996959374287116, −10.26566663470501433905528275676, −9.199490149160226089300331617723, −8.951831489310309906131131164331, −6.46897811046509974709422032178, −5.39036932726915542078513831220, −4.07368242748784542280341370512, −2.62956021753950962986863663502, −0.828647015585665386227358948347,
3.53162341026785044440835747748, 5.14863132619665607110465250884, 6.04880421981961916380514336830, 6.90180812527853486290388648430, 7.58466453865335902796012148906, 9.283059399074271452184741028207, 9.897933262646204038214542027168, 11.30744511173749565740161164838, 12.54632626071949442826357498863, 13.81492203421520540173852286088
