L(s) = 1 | − 23.6i·3-s − 1.38i·5-s − 160.·7-s − 314.·9-s − 129. i·11-s − 759. i·13-s − 32.7·15-s + 323.·17-s + 198. i·19-s + 3.79e3i·21-s + 1.19e3·23-s + 3.12e3·25-s + 1.68e3i·27-s − 5.98e3i·29-s + 4.87e3·31-s + ⋯ |
L(s) = 1 | − 1.51i·3-s − 0.0247i·5-s − 1.23·7-s − 1.29·9-s − 0.321i·11-s − 1.24i·13-s − 0.0375·15-s + 0.271·17-s + 0.126i·19-s + 1.87i·21-s + 0.470·23-s + 0.999·25-s + 0.445i·27-s − 1.32i·29-s + 0.910·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.843 + 0.537i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.843 + 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.305420 - 1.04853i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.305420 - 1.04853i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + 23.6iT - 243T^{2} \) |
| 5 | \( 1 + 1.38iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 160.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 129. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 759. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 323.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 198. iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 1.19e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.98e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 4.87e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.69e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.04e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.87e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 6.29e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.17e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 3.35e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 4.85e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 3.31e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 5.94e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.12e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.37e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.16e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 1.06e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.25e4T + 8.58e9T^{2} \) |
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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
−15.25546180711508807020401445279, −13.59052283237705105510406902952, −12.94044321507329594813167854550, −11.95292660248357466358564265180, −10.17538672952676627118221845070, −8.406398455997316278390177555674, −7.07546570286264314897418744525, −5.92438027993361445072692928814, −2.91646349606784807646718890878, −0.67708065842039853067032152930,
3.32846712608163654962922153318, 4.79240772549305265314692755971, 6.66074353373509311101019341913, 9.016836970479024740168869902060, 9.810033824693913932089497285715, 10.96726838034995488869046341602, 12.53225818050057613116747045949, 14.12325512760850128533687408734, 15.30561546950451144581146187495, 16.25117856790167849318323723709