L(s) = 1 | − 7·5-s + 8.66i·7-s + 8.66i·11-s − 20·13-s − 8·17-s + 10.3i·19-s + 3.46i·23-s + 24·25-s − 10·29-s + 53.6i·31-s − 60.6i·35-s + 10·37-s − 50·41-s + 17.3i·43-s − 86.6i·47-s + ⋯ |
L(s) = 1 | − 1.40·5-s + 1.23i·7-s + 0.787i·11-s − 1.53·13-s − 0.470·17-s + 0.546i·19-s + 0.150i·23-s + 0.959·25-s − 0.344·29-s + 1.73i·31-s − 1.73i·35-s + 0.270·37-s − 1.21·41-s + 0.402i·43-s − 1.84i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.07023379382\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07023379382\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 7T + 25T^{2} \) |
| 7 | \( 1 - 8.66iT - 49T^{2} \) |
| 11 | \( 1 - 8.66iT - 121T^{2} \) |
| 13 | \( 1 + 20T + 169T^{2} \) |
| 17 | \( 1 + 8T + 289T^{2} \) |
| 19 | \( 1 - 10.3iT - 361T^{2} \) |
| 23 | \( 1 - 3.46iT - 529T^{2} \) |
| 29 | \( 1 + 10T + 841T^{2} \) |
| 31 | \( 1 - 53.6iT - 961T^{2} \) |
| 37 | \( 1 - 10T + 1.36e3T^{2} \) |
| 41 | \( 1 + 50T + 1.68e3T^{2} \) |
| 43 | \( 1 - 17.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 86.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 47T + 2.80e3T^{2} \) |
| 59 | \( 1 + 34.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 64T + 3.72e3T^{2} \) |
| 67 | \( 1 - 86.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 55T + 5.32e3T^{2} \) |
| 79 | \( 1 + 6.92iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 29.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 10T + 7.92e3T^{2} \) |
| 97 | \( 1 + 25T + 9.40e3T^{2} \) |
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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
−8.750869417149789313456039066852, −8.246176901344315328549176756328, −7.28022034892751155659984537695, −6.84539612945579376432838782420, −5.45823189088547699354222941324, −4.85623292066934452810044892230, −3.92154845573100617901051522269, −2.88148872859654995614706147326, −1.91934529446445515988839873302, −0.02740116284556155819375141435,
0.72357200160197417824254408195, 2.49917071637595832211077539914, 3.60545285132478823781339792944, 4.26606016885905659939015195923, 4.98225292590337556317854239189, 6.28784452918444716608593250939, 7.31631793353207256452758040323, 7.53596225751821782612953760322, 8.371193506327609195962761832737, 9.323739097780697821974062116186