| L(s) = 1 | + (2.20 − 0.115i)2-s + (−1.65 + 2.04i)3-s + (2.86 − 0.300i)4-s + (0.294 + 2.21i)5-s + (−3.42 + 4.70i)6-s + (−1.02 − 2.43i)7-s + (1.91 − 0.303i)8-s + (−0.819 − 3.85i)9-s + (0.906 + 4.85i)10-s + (4.32 + 0.918i)11-s + (−4.13 + 6.36i)12-s + (1.80 − 0.921i)13-s + (−2.55 − 5.25i)14-s + (−5.02 − 3.07i)15-s + (−1.43 + 0.305i)16-s + (−0.163 + 0.427i)17-s + ⋯ |
| L(s) = 1 | + (1.55 − 0.0817i)2-s + (−0.957 + 1.18i)3-s + (1.43 − 0.150i)4-s + (0.131 + 0.991i)5-s + (−1.39 + 1.92i)6-s + (−0.388 − 0.921i)7-s + (0.677 − 0.107i)8-s + (−0.273 − 1.28i)9-s + (0.286 + 1.53i)10-s + (1.30 + 0.277i)11-s + (−1.19 + 1.83i)12-s + (0.501 − 0.255i)13-s + (−0.681 − 1.40i)14-s + (−1.29 − 0.793i)15-s + (−0.359 + 0.0764i)16-s + (−0.0397 + 0.103i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.65718 + 0.974191i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.65718 + 0.974191i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.294 - 2.21i)T \) |
| 7 | \( 1 + (1.02 + 2.43i)T \) |
| good | 2 | \( 1 + (-2.20 + 0.115i)T + (1.98 - 0.209i)T^{2} \) |
| 3 | \( 1 + (1.65 - 2.04i)T + (-0.623 - 2.93i)T^{2} \) |
| 11 | \( 1 + (-4.32 - 0.918i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-1.80 + 0.921i)T + (7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (0.163 - 0.427i)T + (-12.6 - 11.3i)T^{2} \) |
| 19 | \( 1 + (-0.752 + 7.15i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (0.214 + 4.09i)T + (-22.8 + 2.40i)T^{2} \) |
| 29 | \( 1 + (-4.82 - 6.64i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.349 - 0.784i)T + (-20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (5.20 + 3.37i)T + (15.0 + 33.8i)T^{2} \) |
| 41 | \( 1 + (-0.382 - 0.124i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (1.61 - 1.61i)T - 43iT^{2} \) |
| 47 | \( 1 + (8.18 - 3.14i)T + (34.9 - 31.4i)T^{2} \) |
| 53 | \( 1 + (-0.148 - 0.119i)T + (11.0 + 51.8i)T^{2} \) |
| 59 | \( 1 + (2.29 + 2.55i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (0.888 + 0.800i)T + (6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (-4.00 - 1.53i)T + (49.7 + 44.8i)T^{2} \) |
| 71 | \( 1 + (-0.544 + 0.395i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-9.23 - 14.2i)T + (-29.6 + 66.6i)T^{2} \) |
| 79 | \( 1 + (-5.67 + 12.7i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (0.785 + 4.96i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (-0.0463 + 0.0515i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (-0.650 + 4.10i)T + (-92.2 - 29.9i)T^{2} \) |
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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
−12.93332777202901160004004975579, −11.77115442019766357546797460028, −11.03146825029683138647776981782, −10.44172863310057013120559959756, −9.264488491709079093795842232014, −6.80240129204932883019994407647, −6.42087827251298531835786965230, −5.03357318276450429087243935580, −4.11513958604557426680322139313, −3.22748514881641655806478085376,
1.68051770288351105253500371869, 3.77319884725817938126412692706, 5.24689942097444462477590327241, 6.04025807680433186176604756491, 6.57083129440309991402871742510, 8.203519623705533733191926190539, 9.501072555511154065552285111110, 11.50386893967335049115086616252, 12.04435190688629480014573553347, 12.38668462928429195270899933240
