| L(s) = 1 | + (1.32 − 2.03i)2-s + (0.732 − 1.90i)3-s + (−1.58 − 3.56i)4-s + (−0.600 + 2.15i)5-s + (−2.92 − 4.02i)6-s + (1.28 + 2.31i)7-s + (−4.57 − 0.724i)8-s + (−0.878 − 0.791i)9-s + (3.59 + 4.07i)10-s + (1.38 + 1.53i)11-s + (−7.97 + 0.417i)12-s + (−4.81 − 2.45i)13-s + (6.41 + 0.439i)14-s + (3.67 + 2.72i)15-s + (−2.30 + 2.55i)16-s + (−3.44 + 4.25i)17-s + ⋯ |
| L(s) = 1 | + (0.935 − 1.44i)2-s + (0.423 − 1.10i)3-s + (−0.794 − 1.78i)4-s + (−0.268 + 0.963i)5-s + (−1.19 − 1.64i)6-s + (0.486 + 0.873i)7-s + (−1.61 − 0.256i)8-s + (−0.292 − 0.263i)9-s + (1.13 + 1.28i)10-s + (0.417 + 0.463i)11-s + (−2.30 + 0.120i)12-s + (−1.33 − 0.679i)13-s + (1.71 + 0.117i)14-s + (0.948 + 0.703i)15-s + (−0.575 + 0.639i)16-s + (−0.835 + 1.03i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.648 + 0.761i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.648 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.799858 - 1.73161i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.799858 - 1.73161i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.600 - 2.15i)T \) |
| 7 | \( 1 + (-1.28 - 2.31i)T \) |
| good | 2 | \( 1 + (-1.32 + 2.03i)T + (-0.813 - 1.82i)T^{2} \) |
| 3 | \( 1 + (-0.732 + 1.90i)T + (-2.22 - 2.00i)T^{2} \) |
| 11 | \( 1 + (-1.38 - 1.53i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (4.81 + 2.45i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (3.44 - 4.25i)T + (-3.53 - 16.6i)T^{2} \) |
| 19 | \( 1 + (-2.12 - 0.944i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (2.35 + 1.52i)T + (9.35 + 21.0i)T^{2} \) |
| 29 | \( 1 + (-3.75 + 5.16i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (7.04 - 0.740i)T + (30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-0.0212 - 0.405i)T + (-36.7 + 3.86i)T^{2} \) |
| 41 | \( 1 + (-8.72 + 2.83i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (3.92 + 3.92i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.08 - 1.68i)T + (9.77 - 45.9i)T^{2} \) |
| 53 | \( 1 + (-3.78 - 1.45i)T + (39.3 + 35.4i)T^{2} \) |
| 59 | \( 1 + (-0.0385 - 0.00818i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (-1.48 - 6.96i)T + (-55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-6.86 - 5.55i)T + (13.9 + 65.5i)T^{2} \) |
| 71 | \( 1 + (-9.94 - 7.22i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (2.80 + 0.147i)T + (72.6 + 7.63i)T^{2} \) |
| 79 | \( 1 + (7.96 + 0.837i)T + (77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (1.44 - 9.13i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-2.35 + 0.499i)T + (81.3 - 36.1i)T^{2} \) |
| 97 | \( 1 + (2.60 + 16.4i)T + (-92.2 + 29.9i)T^{2} \) |
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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
−12.35266046475535041225950451886, −11.74282705054787439233522148849, −10.70014172267473292342558877976, −9.749670131813221941010415767163, −8.191155550456989986699065077035, −7.08736592937781143620776880946, −5.71446211365925652570018123650, −4.23607485842064448675995818004, −2.67479386218056523401323380249, −2.00222061792434533325931847834,
3.68881286454260505551771579938, 4.61324134999192580577706078834, 5.08446801941241790711524966640, 6.84711716394702211101430784689, 7.74143588817487301149358866256, 8.894901409010182054500144974619, 9.647925627037030781023715841742, 11.26447648905417396151245880165, 12.41014381513500129681441653314, 13.52132615037781611854343477144
