| L(s) = 1 | + (0.959 − 0.281i)2-s + (−0.468 − 0.540i)3-s + (0.841 − 0.540i)4-s + (−0.317 − 2.20i)5-s + (−0.601 − 0.386i)6-s + (−0.415 + 0.909i)7-s + (0.654 − 0.755i)8-s + (0.354 − 2.46i)9-s + (−0.925 − 2.02i)10-s + (0.232 + 0.0682i)11-s + (−0.686 − 0.201i)12-s + (−0.496 − 1.08i)13-s + (−0.142 + 0.989i)14-s + (−1.04 + 1.20i)15-s + (0.415 − 0.909i)16-s + (−1.30 − 0.840i)17-s + ⋯ |
| L(s) = 1 | + (0.678 − 0.199i)2-s + (−0.270 − 0.312i)3-s + (0.420 − 0.270i)4-s + (−0.141 − 0.986i)5-s + (−0.245 − 0.157i)6-s + (−0.157 + 0.343i)7-s + (0.231 − 0.267i)8-s + (0.118 − 0.820i)9-s + (−0.292 − 0.641i)10-s + (0.0701 + 0.0205i)11-s + (−0.198 − 0.0581i)12-s + (−0.137 − 0.301i)13-s + (−0.0380 + 0.264i)14-s + (−0.269 + 0.311i)15-s + (0.103 − 0.227i)16-s + (−0.317 − 0.203i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0287 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0287 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.20406 - 1.16998i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20406 - 1.16998i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.959 + 0.281i)T \) |
| 7 | \( 1 + (0.415 - 0.909i)T \) |
| 23 | \( 1 + (-4.78 + 0.248i)T \) |
| good | 3 | \( 1 + (0.468 + 0.540i)T + (-0.426 + 2.96i)T^{2} \) |
| 5 | \( 1 + (0.317 + 2.20i)T + (-4.79 + 1.40i)T^{2} \) |
| 11 | \( 1 + (-0.232 - 0.0682i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (0.496 + 1.08i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (1.30 + 0.840i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-0.381 + 0.245i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-2.55 - 1.64i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (2.64 - 3.04i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (0.619 - 4.30i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (0.325 + 2.26i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-8.16 - 9.42i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 8.77T + 47T^{2} \) |
| 53 | \( 1 + (2.10 - 4.60i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (1.17 + 2.58i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (2.78 - 3.20i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (9.52 - 2.79i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-8.04 + 2.36i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (4.71 - 3.02i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (0.165 + 0.362i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (0.759 - 5.27i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (-9.66 - 11.1i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (2.28 + 15.9i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−11.70403928110175523350307067255, −10.71432275039981137685131438994, −9.423682411360253926439122710138, −8.731108724442176366273691170487, −7.35408690716228339842488912585, −6.34352032000008915996175432781, −5.32416388085577415709845263639, −4.36645935709069772426800191112, −2.97441609031470376837844972085, −1.10726228578672800443094734959,
2.41064167082261804741826448711, 3.71858788560951833483373546004, 4.77958674479856793903033351834, 5.93427739508772066654588903170, 7.00501792849494380793285125214, 7.65311090062796308203112570219, 9.093775402676380518053462336512, 10.45051613076026773393815114738, 10.85301316634758155249183020573, 11.74491808010236155336586893687
