L(s) = 1 | − 0.631i·2-s + 2.43·3-s + 1.60·4-s − 2.68i·5-s − 1.53i·6-s − 2.27i·8-s + 2.94·9-s − 1.69·10-s − 4.31·11-s + 3.90·12-s + 5.99·13-s − 6.55i·15-s + 1.76·16-s + 7.07i·17-s − 1.85i·18-s + (−4.32 + 0.579i)19-s + ⋯ |
L(s) = 1 | − 0.446i·2-s + 1.40·3-s + 0.800·4-s − 1.20i·5-s − 0.628i·6-s − 0.804i·8-s + 0.981·9-s − 0.537·10-s − 1.30·11-s + 1.12·12-s + 1.66·13-s − 1.69i·15-s + 0.441·16-s + 1.71i·17-s − 0.438i·18-s + (−0.991 + 0.133i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.42104 - 1.80290i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.42104 - 1.80290i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 19 | \( 1 + (4.32 - 0.579i)T \) |
good | 2 | \( 1 + 0.631iT - 2T^{2} \) |
| 3 | \( 1 - 2.43T + 3T^{2} \) |
| 5 | \( 1 + 2.68iT - 5T^{2} \) |
| 11 | \( 1 + 4.31T + 11T^{2} \) |
| 13 | \( 1 - 5.99T + 13T^{2} \) |
| 17 | \( 1 - 7.07iT - 17T^{2} \) |
| 23 | \( 1 - 4.78T + 23T^{2} \) |
| 29 | \( 1 + 5.13iT - 29T^{2} \) |
| 31 | \( 1 + 3.37T + 31T^{2} \) |
| 37 | \( 1 - 1.12iT - 37T^{2} \) |
| 41 | \( 1 + 6.58T + 41T^{2} \) |
| 43 | \( 1 + 1.54T + 43T^{2} \) |
| 47 | \( 1 - 4.63iT - 47T^{2} \) |
| 53 | \( 1 + 8.82iT - 53T^{2} \) |
| 59 | \( 1 - 0.570T + 59T^{2} \) |
| 61 | \( 1 - 1.54iT - 61T^{2} \) |
| 67 | \( 1 - 12.9iT - 67T^{2} \) |
| 71 | \( 1 + 5.14iT - 71T^{2} \) |
| 73 | \( 1 - 12.9iT - 73T^{2} \) |
| 79 | \( 1 + 0.719iT - 79T^{2} \) |
| 83 | \( 1 - 12.1iT - 83T^{2} \) |
| 89 | \( 1 - 0.782T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−9.929536470318840470679592433682, −8.726923155951440375067174998306, −8.446016104434032802844656221776, −7.77581576156678828348409585988, −6.49574328800758415212091120135, −5.53898440086850749298121052244, −4.13214548252883626126716017655, −3.39836103491176564614650179642, −2.26411082319031846559891759994, −1.36628347729678108424067272994,
2.02570012919373084067581628972, 2.95982418350263203815309871213, 3.34061801527791313770146112969, 5.07128579710796848694728380400, 6.23424132774932022840554324519, 7.08017700162195736597843284466, 7.57906391924462546747199614646, 8.473732620458818142278611868462, 9.114195027059344745993447461620, 10.44800123663475792089178035936