| L(s) = 1 | + (0.0713 − 0.997i)2-s + (0.977 − 0.212i)3-s + (−0.989 − 0.142i)4-s + (−1.72 − 1.41i)5-s + (−0.142 − 0.989i)6-s + (−4.29 + 2.34i)7-s + (−0.212 + 0.977i)8-s + (0.909 − 0.415i)9-s + (−1.53 + 1.62i)10-s + (1.50 + 1.30i)11-s + (−0.997 + 0.0713i)12-s + (−0.133 + 0.244i)13-s + (2.03 + 4.44i)14-s + (−1.99 − 1.01i)15-s + (0.959 + 0.281i)16-s + (−2.43 − 1.82i)17-s + ⋯ |
| L(s) = 1 | + (0.0504 − 0.705i)2-s + (0.564 − 0.122i)3-s + (−0.494 − 0.0711i)4-s + (−0.773 − 0.634i)5-s + (−0.0580 − 0.404i)6-s + (−1.62 + 0.885i)7-s + (−0.0751 + 0.345i)8-s + (0.303 − 0.138i)9-s + (−0.486 + 0.513i)10-s + (0.454 + 0.393i)11-s + (−0.287 + 0.0205i)12-s + (−0.0370 + 0.0679i)13-s + (0.542 + 1.18i)14-s + (−0.513 − 0.263i)15-s + (0.239 + 0.0704i)16-s + (−0.590 − 0.442i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0656 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0656 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.320245 + 0.299863i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.320245 + 0.299863i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0713 + 0.997i)T \) |
| 3 | \( 1 + (-0.977 + 0.212i)T \) |
| 5 | \( 1 + (1.72 + 1.41i)T \) |
| 23 | \( 1 + (-0.205 - 4.79i)T \) |
| good | 7 | \( 1 + (4.29 - 2.34i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (-1.50 - 1.30i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.133 - 0.244i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (2.43 + 1.82i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (0.800 - 5.56i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-1.36 + 0.195i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (4.02 - 2.58i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (8.75 + 3.26i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (3.49 - 7.65i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (0.412 + 1.89i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (-1.51 - 1.51i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.56 + 4.69i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (-1.00 - 3.43i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (8.04 + 12.5i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (14.5 + 1.04i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (2.87 + 3.32i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-2.88 - 3.85i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (-7.25 + 2.12i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-1.54 + 4.13i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (-1.70 - 1.09i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-4.74 - 12.7i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−10.59661483824633044734380572946, −9.524706494668633500834645569447, −9.204064459563790627019470830266, −8.363440232606963432459197190117, −7.26957586341745348258898321180, −6.22533919812160690038759519096, −5.04564752603974651798395284108, −3.77179801655543751894224875831, −3.20221925918410342064622085947, −1.79142742854715707077716562523,
0.20601422094040869909917200479, 2.88859723119282991832244307269, 3.70398362858205811332603227198, 4.47905991411240839712640377696, 6.18368709279822779144348774224, 6.87378737212981831762845285041, 7.35460031590410102259932406205, 8.591524318567127796976121795423, 9.166954228648810412545778373105, 10.29425052871555038132896138974
