L(s) = 1 | − 0.317i·5-s − i·7-s − 1.41·11-s − 4.44·13-s − 5.97i·17-s − 2.44i·19-s + 5.65·23-s + 4.89·25-s + 5.51i·29-s + 6.44i·31-s − 0.317·35-s + 9·37-s + 7.24i·41-s − 6.34i·43-s + 10.8·47-s + ⋯ |
L(s) = 1 | − 0.142i·5-s − 0.377i·7-s − 0.426·11-s − 1.23·13-s − 1.44i·17-s − 0.561i·19-s + 1.17·23-s + 0.979·25-s + 1.02i·29-s + 1.15i·31-s − 0.0537·35-s + 1.47·37-s + 1.13i·41-s − 0.968i·43-s + 1.58·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.015344525\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.015344525\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + 0.317iT - 5T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 + 4.44T + 13T^{2} \) |
| 17 | \( 1 + 5.97iT - 17T^{2} \) |
| 19 | \( 1 + 2.44iT - 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 - 5.51iT - 29T^{2} \) |
| 31 | \( 1 - 6.44iT - 31T^{2} \) |
| 37 | \( 1 - 9T + 37T^{2} \) |
| 41 | \( 1 - 7.24iT - 41T^{2} \) |
| 43 | \( 1 + 6.34iT - 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 + 11.9iT - 53T^{2} \) |
| 59 | \( 1 + 5.83T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + 5.79iT - 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 9.79T + 73T^{2} \) |
| 79 | \( 1 + 14.3iT - 79T^{2} \) |
| 83 | \( 1 + 8.02T + 83T^{2} \) |
| 89 | \( 1 - 15.4iT - 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−7.61962157045837616922071999032, −7.11773521575869507755976082858, −6.66964989575803218073464380121, −5.38689188521068343833080086417, −4.95389391468189891353637781172, −4.38517056326215640706808810316, −2.93986100037202406913883720451, −2.80200742545663678989121902775, −1.33541270929710765694681835280, −0.27461725918572087729231263113,
1.19241163964774588089922152663, 2.41956412658744092732424882135, 2.86338300770114560825383493215, 4.11331295231782530461558292651, 4.60778296023770561200094034392, 5.71007051929001492565790794333, 5.99165718985852923205578687662, 7.04695972018720291151006690361, 7.64539829460196804976377160114, 8.230054114844299567113760346207