L(s) = 1 | − 21.9i·3-s + 184. i·5-s − 1.05e3·7-s + 1.70e3·9-s + 4.32e3i·11-s + 1.12e4i·13-s + 4.05e3·15-s − 2.17e4·17-s + 4.54e4i·19-s + 2.30e4i·21-s − 4.41e3·23-s + 4.39e4·25-s − 8.54e4i·27-s − 2.36e4i·29-s − 7.29e4·31-s + ⋯ |
L(s) = 1 | − 0.469i·3-s + 0.661i·5-s − 1.15·7-s + 0.779·9-s + 0.979i·11-s + 1.42i·13-s + 0.310·15-s − 1.07·17-s + 1.52i·19-s + 0.543i·21-s − 0.0756·23-s + 0.562·25-s − 0.835i·27-s − 0.180i·29-s − 0.439·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.137 - 0.990i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.137 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.727854 + 0.835600i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.727854 + 0.835600i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + 21.9iT - 2.18e3T^{2} \) |
| 5 | \( 1 - 184. iT - 7.81e4T^{2} \) |
| 7 | \( 1 + 1.05e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 4.32e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 1.12e4iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 2.17e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.54e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 4.41e3T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.36e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 7.29e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.83e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 4.11e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 9.61e4iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 1.56e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 6.86e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 1.79e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 - 1.36e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 1.08e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 5.60e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.16e4T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.34e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.82e5iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 1.34e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 7.32e6T + 8.07e13T^{2} \) |
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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
−15.61601200075383968768194692901, −14.32227447456084385506275002719, −13.05357997620429506881090551382, −12.09809966957710376484747233310, −10.41954573956791887129760552716, −9.310780730292977823859471632986, −7.23449504136212369919170148786, −6.46629586391283543646346646067, −4.03800118799020079338147600589, −2.01549198704809618541833142767,
0.51457261522676747015917469618, 3.23214193338561395400116646262, 4.97523076659672317495670699754, 6.67674772481399100335420585473, 8.582337220508402254414083829671, 9.748027928008219967675131228300, 10.97934189533686084441418889251, 12.80498828864842461720506794851, 13.33737476709256844314061337516, 15.38442755022061614886999782615