| L(s) = 1 | + (−1.68 − 0.306i)2-s + (3.00 − 0.828i)3-s + (0.879 + 0.330i)4-s + (−0.395 + 1.21i)5-s + (−5.31 + 0.478i)6-s + (−1.19 − 2.22i)7-s + (1.56 + 0.932i)8-s + (5.74 − 3.43i)9-s + (1.03 − 1.93i)10-s + (−1.22 + 2.85i)11-s + (2.91 + 0.262i)12-s + (−1.09 − 2.56i)13-s + (1.33 + 4.11i)14-s + (−0.178 + 3.97i)15-s + (−3.76 − 3.28i)16-s + (−4.83 + 3.51i)17-s + ⋯ |
| L(s) = 1 | + (−1.19 − 0.216i)2-s + (1.73 − 0.478i)3-s + (0.439 + 0.165i)4-s + (−0.176 + 0.544i)5-s + (−2.17 + 0.195i)6-s + (−0.452 − 0.840i)7-s + (0.551 + 0.329i)8-s + (1.91 − 1.14i)9-s + (0.328 − 0.610i)10-s + (−0.368 + 0.861i)11-s + (0.841 + 0.0757i)12-s + (−0.303 − 0.710i)13-s + (0.357 + 1.09i)14-s + (−0.0461 + 1.02i)15-s + (−0.940 − 0.821i)16-s + (−1.17 + 0.852i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.746275 - 0.226904i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.746275 - 0.226904i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 + (6.39 + 5.48i)T \) |
| good | 2 | \( 1 + (1.68 + 0.306i)T + (1.87 + 0.702i)T^{2} \) |
| 3 | \( 1 + (-3.00 + 0.828i)T + (2.57 - 1.53i)T^{2} \) |
| 5 | \( 1 + (0.395 - 1.21i)T + (-4.04 - 2.93i)T^{2} \) |
| 7 | \( 1 + (1.19 + 2.22i)T + (-3.85 + 5.84i)T^{2} \) |
| 11 | \( 1 + (1.22 - 2.85i)T + (-7.60 - 7.95i)T^{2} \) |
| 13 | \( 1 + (1.09 + 2.56i)T + (-8.98 + 9.39i)T^{2} \) |
| 17 | \( 1 + (4.83 - 3.51i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.268 - 5.97i)T + (-18.9 + 1.70i)T^{2} \) |
| 23 | \( 1 + (0.313 + 0.150i)T + (14.3 + 17.9i)T^{2} \) |
| 29 | \( 1 + (3.28 + 4.97i)T + (-11.3 + 26.6i)T^{2} \) |
| 31 | \( 1 + (4.28 - 3.73i)T + (4.16 - 30.7i)T^{2} \) |
| 37 | \( 1 + (-4.07 + 1.96i)T + (23.0 - 28.9i)T^{2} \) |
| 41 | \( 1 + (-0.557 + 2.44i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (1.17 + 8.69i)T + (-41.4 + 11.4i)T^{2} \) |
| 47 | \( 1 + (-2.92 - 0.808i)T + (40.3 + 24.1i)T^{2} \) |
| 53 | \( 1 + (-9.80 + 3.67i)T + (39.9 - 34.8i)T^{2} \) |
| 59 | \( 1 + (6.98 + 0.628i)T + (58.0 + 10.5i)T^{2} \) |
| 61 | \( 1 + (-0.503 + 0.936i)T + (-33.6 - 50.9i)T^{2} \) |
| 67 | \( 1 + (-3.10 - 1.16i)T + (50.4 + 44.0i)T^{2} \) |
| 73 | \( 1 + (-9.79 - 1.77i)T + (68.3 + 25.6i)T^{2} \) |
| 79 | \( 1 + (3.39 + 2.02i)T + (37.4 + 69.5i)T^{2} \) |
| 83 | \( 1 + (0.0918 + 0.00826i)T + (81.6 + 14.8i)T^{2} \) |
| 89 | \( 1 + (4.08 - 1.53i)T + (67.0 - 58.5i)T^{2} \) |
| 97 | \( 1 + (2.32 + 10.1i)T + (-87.3 + 42.0i)T^{2} \) |
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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
−14.59975997024389348030483353080, −13.55241965577034044059371550528, −12.68196033857021957953221951972, −10.59092937210950540252714456141, −9.918931655021039928131047841319, −8.797904960313708903607732174927, −7.74747484363232283909193839765, −7.11501703657230346349641378620, −3.83448097771935955988333219525, −2.10986746170971843923125849163,
2.63497387219329658279456496839, 4.51335498237861006222334552332, 7.12121604343247652308580416994, 8.375308427332005878668478007312, 9.093228918505053086786012969448, 9.443356935454459817491515545307, 11.04589933108916343047287756775, 12.99062086678387462657254845088, 13.70987141645220963622919177081, 15.08278301726148680814560718171
