L(s) = 1 | + 1.41i·2-s − 1.23·3-s − 2.00·4-s − 2.23·5-s − 1.74i·6-s − 2.49i·7-s − 2.82i·8-s − 7.47·9-s − 3.16i·10-s + 2.47·12-s − 1.08i·13-s + 3.52·14-s + 2.76·15-s + 4.00·16-s − 10.9i·17-s − 10.5i·18-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.412·3-s − 0.500·4-s − 0.447·5-s − 0.291i·6-s − 0.356i·7-s − 0.353i·8-s − 0.830·9-s − 0.316i·10-s + 0.206·12-s − 0.0831i·13-s + 0.251·14-s + 0.184·15-s + 0.250·16-s − 0.641i·17-s − 0.587i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.426 - 0.904i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7715165317\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7715165317\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 5 | \( 1 + 2.23T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 1.23T + 9T^{2} \) |
| 7 | \( 1 + 2.49iT - 49T^{2} \) |
| 13 | \( 1 + 1.08iT - 169T^{2} \) |
| 17 | \( 1 + 10.9iT - 289T^{2} \) |
| 19 | \( 1 + 7.14iT - 361T^{2} \) |
| 23 | \( 1 - 26.0T + 529T^{2} \) |
| 29 | \( 1 + 7.73iT - 841T^{2} \) |
| 31 | \( 1 + 38.8T + 961T^{2} \) |
| 37 | \( 1 + 32.1T + 1.36e3T^{2} \) |
| 41 | \( 1 - 17.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 37.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 20.4T + 2.20e3T^{2} \) |
| 53 | \( 1 + 44.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + 2.47T + 3.48e3T^{2} \) |
| 61 | \( 1 - 38.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 129.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 48.9T + 5.04e3T^{2} \) |
| 73 | \( 1 - 74.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 67.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 102. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 113.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 123.T + 9.40e3T^{2} \) |
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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
−9.603583616823268002770272683212, −8.889564981580533744761426432651, −8.098166512090335857926334554628, −7.22608776787412226236995947564, −6.61464976125120406723121437803, −5.53181861463452434175481160660, −4.94155612952551053835409730360, −3.84068910556258859510452083486, −2.76881156851982997677110075543, −0.847602517176647854752201752423,
0.33323116979270445754903863040, 1.81324435535522432848723530905, 3.03587048664586416017665096190, 3.86062149079083498944832279224, 5.05314949698846288149797361938, 5.68515341915116314224423064342, 6.75201947306110184713602119556, 7.78479208509240259826810148517, 8.737838176992501707950073209795, 9.144130089567479217717030040813