L(s) = 1 | − 4i·2-s − i·3-s − 16·4-s − 4·6-s − 166i·7-s + 64i·8-s + 242·9-s − 121·11-s + 16i·12-s − 692i·13-s − 664·14-s + 256·16-s − 738i·17-s − 968i·18-s − 1.42e3·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.0641i·3-s − 0.5·4-s − 0.0453·6-s − 1.28i·7-s + 0.353i·8-s + 0.995·9-s − 0.301·11-s + 0.0320i·12-s − 1.13i·13-s − 0.905·14-s + 0.250·16-s − 0.619i·17-s − 0.704i·18-s − 0.904·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.423973614\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.423973614\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 4iT \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
good | 3 | \( 1 + iT - 243T^{2} \) |
| 7 | \( 1 + 166iT - 1.68e4T^{2} \) |
| 13 | \( 1 + 692iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 738iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.42e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.77e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 2.06e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.24e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.47e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 5.30e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.77e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.71e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 3.07e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 3.49e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.59e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.53e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 1.33e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.32e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 7.72e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.50e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 1.25e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.88e4iT - 8.58e9T^{2} \) |
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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
−9.902828077097590753236647029353, −8.690529889178830796639051056589, −7.62282010702504340294586630868, −7.03989666314006893544576883718, −5.62666795124047377613043516883, −4.47206890521511149385129833377, −3.76786314485316018272266988607, −2.52412045818749208559136438371, −1.17284060619214362295062690224, −0.34790198358547525806986581996,
1.47190582118652475489307462558, 2.67726972979210351509070079934, 4.24859334485509548506669224700, 4.89541214036486517515099927716, 6.24525504963952091761986971035, 6.61906687308860739010336821652, 7.973086720598388884943927681493, 8.616530177468983034762567960627, 9.503289496218227891152006339862, 10.26772840484136251331668946053