| L(s) = 1 | − 12.7·2-s − 348.·4-s − 2.01e3·5-s − 665.·7-s + 1.10e4·8-s + 2.58e4·10-s − 8.54e4·11-s + 8.48e4·13-s + 8.52e3·14-s + 3.73e4·16-s + 1.39e5·17-s + 9.93e5·19-s + 7.03e5·20-s + 1.09e6·22-s − 7.58e5·23-s + 2.12e6·25-s − 1.08e6·26-s + 2.31e5·28-s + 4.04e6·29-s − 3.01e6·31-s − 6.11e6·32-s − 1.78e6·34-s + 1.34e6·35-s + 2.04e6·37-s − 1.27e7·38-s − 2.22e7·40-s + 7.85e6·41-s + ⋯ |
| L(s) = 1 | − 0.565·2-s − 0.680·4-s − 1.44·5-s − 0.104·7-s + 0.950·8-s + 0.817·10-s − 1.75·11-s + 0.823·13-s + 0.0592·14-s + 0.142·16-s + 0.405·17-s + 1.74·19-s + 0.982·20-s + 0.995·22-s − 0.565·23-s + 1.08·25-s − 0.465·26-s + 0.0712·28-s + 1.06·29-s − 0.586·31-s − 1.03·32-s − 0.229·34-s + 0.151·35-s + 0.179·37-s − 0.988·38-s − 1.37·40-s + 0.433·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.6016990454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6016990454\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 12.7T + 512T^{2} \) |
| 5 | \( 1 + 2.01e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 665.T + 4.03e7T^{2} \) |
| 11 | \( 1 + 8.54e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 8.48e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.39e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 9.93e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 7.58e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.04e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.01e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 2.04e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 7.85e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.72e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.29e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.02e8T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.15e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.29e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.95e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.18e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 4.41e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.09e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.86e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 7.64e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 8.15e5T + 7.60e17T^{2} \) |
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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
−15.67173097663781139944863050371, −13.94941583943653369722181567876, −12.70549214359674256862552068138, −11.25446992254419291436897252033, −9.967254151733503670671170761804, −8.296357636722416581208824487911, −7.62188148666645997054177371824, −5.07685432850775138028618310258, −3.47997285810073373554924224322, −0.62967216230516064780684076388,
0.62967216230516064780684076388, 3.47997285810073373554924224322, 5.07685432850775138028618310258, 7.62188148666645997054177371824, 8.296357636722416581208824487911, 9.967254151733503670671170761804, 11.25446992254419291436897252033, 12.70549214359674256862552068138, 13.94941583943653369722181567876, 15.67173097663781139944863050371
