L(s) = 1 | + (−4.80 − 1.39i)5-s − 12.3i·7-s + 4.62i·11-s + 1.26i·13-s + 22.9·17-s + 32.8·19-s + 0.797·23-s + (21.1 + 13.3i)25-s + 39.6i·29-s + 1.46·31-s + (−17.2 + 59.4i)35-s − 44.9i·37-s − 39.1i·41-s − 36.3i·43-s + 61.0·47-s + ⋯ |
L(s) = 1 | + (−0.960 − 0.278i)5-s − 1.76i·7-s + 0.420i·11-s + 0.0971i·13-s + 1.34·17-s + 1.73·19-s + 0.0346·23-s + (0.844 + 0.534i)25-s + 1.36i·29-s + 0.0471·31-s + (−0.492 + 1.69i)35-s − 1.21i·37-s − 0.954i·41-s − 0.846i·43-s + 1.29·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.278 + 0.960i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.278 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.535463013\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.535463013\) |
\(L(2)\) |
|
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 | \( 1 + (4.80 + 1.39i)T \) |
good | 7 | \( 1 + 12.3iT - 49T^{2} \) |
| 11 | \( 1 - 4.62iT - 121T^{2} \) |
| 13 | \( 1 - 1.26iT - 169T^{2} \) |
| 17 | \( 1 - 22.9T + 289T^{2} \) |
| 19 | \( 1 - 32.8T + 361T^{2} \) |
| 23 | \( 1 - 0.797T + 529T^{2} \) |
| 29 | \( 1 - 39.6iT - 841T^{2} \) |
| 31 | \( 1 - 1.46T + 961T^{2} \) |
| 37 | \( 1 + 44.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 39.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 36.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 61.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 13.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 58.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 55.0T + 3.72e3T^{2} \) |
| 67 | \( 1 + 66.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 70.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 23.1iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 29.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + 130.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 48.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 19.3iT - 9.40e3T^{2} \) |
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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
−9.012783304415675952245177705187, −7.85947157782261004992899758519, −7.40533724183352375801332843614, −6.99348540285743115477453676375, −5.52025684769387846530935367096, −4.72156286496503206303122153194, −3.76823559236440835852241789872, −3.29767072346966801979990488776, −1.36601909579143001084031837267, −0.51964260646965960530737279464,
1.12663787707818440094352691697, 2.78075784958908189596068245153, 3.17809237978284322984956921858, 4.47047987755116088317513289929, 5.53393696217062530794608356684, 5.97317949754081594186191197393, 7.18957562080760302773979446460, 7.995697848362464508072734821892, 8.476673864858106023870330417573, 9.460087340845937019393774021233