L(s) = 1 | − 7-s − 4.24i·11-s − 7·13-s + 4.24i·17-s − 7·19-s + 29.6i·23-s + 29.6i·29-s + 17·31-s − 16·37-s − 50.9i·41-s − 55·43-s + 46.6i·47-s − 48·49-s + 84.8i·53-s + 55.1i·59-s + ⋯ |
L(s) = 1 | − 0.142·7-s − 0.385i·11-s − 0.538·13-s + 0.249i·17-s − 0.368·19-s + 1.29i·23-s + 1.02i·29-s + 0.548·31-s − 0.432·37-s − 1.24i·41-s − 1.27·43-s + 0.992i·47-s − 0.979·49-s + 1.60i·53-s + 0.934i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7960827985\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7960827985\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + T + 49T^{2} \) |
| 11 | \( 1 + 4.24iT - 121T^{2} \) |
| 13 | \( 1 + 7T + 169T^{2} \) |
| 17 | \( 1 - 4.24iT - 289T^{2} \) |
| 19 | \( 1 + 7T + 361T^{2} \) |
| 23 | \( 1 - 29.6iT - 529T^{2} \) |
| 29 | \( 1 - 29.6iT - 841T^{2} \) |
| 31 | \( 1 - 17T + 961T^{2} \) |
| 37 | \( 1 + 16T + 1.36e3T^{2} \) |
| 41 | \( 1 + 50.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 55T + 1.84e3T^{2} \) |
| 47 | \( 1 - 46.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 84.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 55.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 65T + 3.72e3T^{2} \) |
| 67 | \( 1 + 49T + 4.48e3T^{2} \) |
| 71 | \( 1 + 50.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 88T + 5.32e3T^{2} \) |
| 79 | \( 1 + 40T + 6.24e3T^{2} \) |
| 83 | \( 1 - 156. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 101. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 41T + 9.40e3T^{2} \) |
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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
−10.21905658310920424624800611829, −9.351530177106777895322213775939, −8.583328510590130601161354482856, −7.64503409001345623449333398466, −6.83270660119798512929282084297, −5.84316269493855837198368425176, −4.97677967019516916225446648597, −3.83394919632874867023228838620, −2.82307665051756833932936444782, −1.44155157006035149366648738820,
0.25413368709989913992283162752, 1.95701962426430871495074973759, 3.05219076596568358860490604859, 4.32692976011623481158418626891, 5.08554236692868200805494491873, 6.30727035105721454465504149852, 6.95618777685981261028276547107, 8.026068277993017347704229638670, 8.698942946515618136103189578833, 9.873532653723532485609668440358