L(s) = 1 | + (−2.11 − 1.22i)2-s + (0.0609 − 0.105i)3-s + (1.98 + 3.44i)4-s + (−0.257 + 0.148i)6-s + (−0.358 + 0.207i)7-s − 4.82i·8-s + (1.49 + 2.58i)9-s + (−3.97 − 2.29i)11-s + 0.484·12-s + (−2.36 + 2.72i)13-s + 1.01·14-s + (−1.91 + 3.32i)16-s + (−0.372 − 0.644i)17-s − 7.29i·18-s + (−6.47 + 3.73i)19-s + ⋯ |
L(s) = 1 | + (−1.49 − 0.864i)2-s + (0.0351 − 0.0609i)3-s + (0.993 + 1.72i)4-s + (−0.105 + 0.0607i)6-s + (−0.135 + 0.0782i)7-s − 1.70i·8-s + (0.497 + 0.861i)9-s + (−1.19 − 0.691i)11-s + 0.139·12-s + (−0.655 + 0.755i)13-s + 0.270·14-s + (−0.479 + 0.831i)16-s + (−0.0902 − 0.156i)17-s − 1.71i·18-s + (−1.48 + 0.857i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.227436 + 0.190152i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.227436 + 0.190152i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 + (2.36 - 2.72i)T \) |
good | 2 | \( 1 + (2.11 + 1.22i)T + (1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.0609 + 0.105i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (0.358 - 0.207i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3.97 + 2.29i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (0.372 + 0.644i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (6.47 - 3.73i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.17 - 5.50i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.80 + 4.85i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 3.60iT - 31T^{2} \) |
| 37 | \( 1 + (-5.57 - 3.21i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.51 - 2.60i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.96 - 8.60i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 1.52iT - 47T^{2} \) |
| 53 | \( 1 + 7.56T + 53T^{2} \) |
| 59 | \( 1 + (3.15 - 1.82i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.95 + 10.3i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.67 + 3.27i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (5.89 - 3.40i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 3.08iT - 73T^{2} \) |
| 79 | \( 1 - 1.13T + 79T^{2} \) |
| 83 | \( 1 + 12.1iT - 83T^{2} \) |
| 89 | \( 1 + (-6.04 - 3.48i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-11.0 + 6.35i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.44802238310324300322688319655, −10.69783884846270544521046198815, −10.02971741095443815712253270785, −9.201644720661352503319775185511, −7.940376027246813789914118343412, −7.78319874416878656609112645102, −6.20604567488643145627661739144, −4.55731415482343977295801542818, −2.85803243941989938867647444984, −1.81356005966812206415007407291,
0.32154789985711386939701456160, 2.38538453759267210300621021716, 4.50087740828243849392083044096, 5.92343468890952929413705692336, 6.87708398928204914620492387348, 7.63655594159426590246184690148, 8.556346246814333452580341713892, 9.427435827710622024393577146147, 10.32297164993288160948684298045, 10.71979644048979074356072472748