| L(s) = 1 | + (−135. + 234. i)5-s + (87.0 + 903. i)7-s + (−2.75e3 − 4.76e3i)11-s + 8.80e3·13-s + (3.42e3 + 5.92e3i)17-s + (1.32e4 − 2.29e4i)19-s + (4.13e4 − 7.15e4i)23-s + (2.27e3 + 3.94e3i)25-s + 2.33e5·29-s + (1.98e4 + 3.43e4i)31-s + (−2.23e5 − 1.02e5i)35-s + (−1.15e5 + 2.00e5i)37-s − 4.03e5·41-s − 1.22e5·43-s + (−5.26e4 + 9.12e4i)47-s + ⋯ |
| L(s) = 1 | + (−0.485 + 0.840i)5-s + (0.0958 + 0.995i)7-s + (−0.623 − 1.08i)11-s + 1.11·13-s + (0.168 + 0.292i)17-s + (0.442 − 0.766i)19-s + (0.708 − 1.22i)23-s + (0.0291 + 0.0504i)25-s + 1.77·29-s + (0.119 + 0.207i)31-s + (−0.883 − 0.402i)35-s + (−0.376 + 0.651i)37-s − 0.915·41-s − 0.235·43-s + (−0.0740 + 0.128i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.933428094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.933428094\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-87.0 - 903. i)T \) |
| good | 5 | \( 1 + (135. - 234. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 11 | \( 1 + (2.75e3 + 4.76e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 - 8.80e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + (-3.42e3 - 5.92e3i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-1.32e4 + 2.29e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-4.13e4 + 7.15e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 - 2.33e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + (-1.98e4 - 3.43e4i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (1.15e5 - 2.00e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + 4.03e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.22e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (5.26e4 - 9.12e4i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-6.65e5 - 1.15e6i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (4.29e5 + 7.43e5i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-5.25e5 + 9.10e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-2.21e6 - 3.83e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 - 1.54e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + (-1.07e6 - 1.86e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (9.54e5 - 1.65e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + 8.56e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-5.54e6 + 9.59e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 - 1.13e6T + 8.07e13T^{2} \) |
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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
−11.05252232851445260462673572315, −10.25430782903754197419512383193, −8.701847080665749170275684298360, −8.340765230918003337959207716916, −6.90135889445242639907630778761, −6.05254530005814241047735116821, −4.91632248061637000441439845118, −3.32641575926004942750310788489, −2.67869537588722024190109286808, −0.910766372001103737422720415968,
0.60147936039658374280973889753, 1.56360838039804702958263333052, 3.37955294112655147833808401235, 4.41432456288309176830312139192, 5.29651685042953776213501277519, 6.80091227658448261601146706792, 7.74412200077521482529078752686, 8.520985315514650236493306919015, 9.754303079740184206598233681788, 10.53965223069208353399254829188
