L(s) = 1 | − 0.680i·5-s − i·7-s − 1.70·11-s + 5.59·13-s + 3.68i·17-s − 4.46i·19-s − 1.16·23-s + 4.53·25-s − 1.77i·29-s + 2.58i·31-s − 0.680·35-s + 1.36·37-s − 1.31i·41-s − 4.71i·43-s + 3.87·47-s + ⋯ |
L(s) = 1 | − 0.304i·5-s − 0.377i·7-s − 0.513·11-s + 1.55·13-s + 0.894i·17-s − 1.02i·19-s − 0.242·23-s + 0.907·25-s − 0.329i·29-s + 0.464i·31-s − 0.115·35-s + 0.223·37-s − 0.205i·41-s − 0.718i·43-s + 0.565·47-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
\(L(1)\) |
\(\approx\) |
\(2.028667621\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.028667621\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.680iT - 5T^{2} \) |
| 11 | \( 1 + 1.70T + 11T^{2} \) |
| 13 | \( 1 - 5.59T + 13T^{2} \) |
| 17 | \( 1 - 3.68iT - 17T^{2} \) |
| 19 | \( 1 + 4.46iT - 19T^{2} \) |
| 23 | \( 1 + 1.16T + 23T^{2} \) |
| 29 | \( 1 + 1.77iT - 29T^{2} \) |
| 31 | \( 1 - 2.58iT - 31T^{2} \) |
| 37 | \( 1 - 1.36T + 37T^{2} \) |
| 41 | \( 1 + 1.31iT - 41T^{2} \) |
| 43 | \( 1 + 4.71iT - 43T^{2} \) |
| 47 | \( 1 - 3.87T + 47T^{2} \) |
| 53 | \( 1 - 6.39iT - 53T^{2} \) |
| 59 | \( 1 - 0.948T + 59T^{2} \) |
| 61 | \( 1 + 1.40T + 61T^{2} \) |
| 67 | \( 1 - 1.28iT - 67T^{2} \) |
| 71 | \( 1 - 3.09T + 71T^{2} \) |
| 73 | \( 1 + 5.79T + 73T^{2} \) |
| 79 | \( 1 + 1.72iT - 79T^{2} \) |
| 83 | \( 1 - 7.69T + 83T^{2} \) |
| 89 | \( 1 - 0.962iT - 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−8.063329238246267115351513967435, −7.29681922056566914484813764161, −6.51994234714341042483467150217, −5.90717042563767409064823284389, −5.10203024039185233904085498511, −4.29483073257915870848780039507, −3.61626361889285964598719170914, −2.71782396818447004851299855948, −1.59676175212266313027200696047, −0.66327007916709642747725611270,
0.910430080580916538077047288434, 2.00023693758714212507218213439, 2.97620546487330044795037745849, 3.61243715586064724818549675784, 4.55039638349690451803034052650, 5.40289550755453513517518778578, 6.04921055341478705961194598676, 6.65466125800075259619059265532, 7.53210609357689987492261295121, 8.171982718320611146971175415274