| L(s) = 1 | + (9.26 − 6.48i)2-s + (−13.5 + 5.62i)3-s + (43.7 − 120. i)4-s + (−202. + 489. i)5-s + (−89.3 + 140. i)6-s + (−1.16e3 − 1.16e3i)7-s + (−374. − 1.39e3i)8-s + (−1.39e3 + 1.39e3i)9-s + (1.29e3 + 5.85e3i)10-s + (−1.98e3 − 822. i)11-s + (82.1 + 1.87e3i)12-s + (3.51e3 + 8.48e3i)13-s + (−1.82e4 − 3.22e3i)14-s − 7.78e3i·15-s + (−1.25e4 − 1.05e4i)16-s − 6.87e3i·17-s + ⋯ |
| L(s) = 1 | + (0.819 − 0.573i)2-s + (−0.290 + 0.120i)3-s + (0.341 − 0.939i)4-s + (−0.725 + 1.75i)5-s + (−0.168 + 0.265i)6-s + (−1.27 − 1.27i)7-s + (−0.258 − 0.965i)8-s + (−0.637 + 0.637i)9-s + (0.410 + 1.85i)10-s + (−0.449 − 0.186i)11-s + (0.0137 + 0.313i)12-s + (0.443 + 1.07i)13-s + (−1.78 − 0.314i)14-s − 0.595i·15-s + (−0.766 − 0.642i)16-s − 0.339i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 - 0.625i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.779 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0890606 + 0.253316i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0890606 + 0.253316i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-9.26 + 6.48i)T \) |
| good | 3 | \( 1 + (13.5 - 5.62i)T + (1.54e3 - 1.54e3i)T^{2} \) |
| 5 | \( 1 + (202. - 489. i)T + (-5.52e4 - 5.52e4i)T^{2} \) |
| 7 | \( 1 + (1.16e3 + 1.16e3i)T + 8.23e5iT^{2} \) |
| 11 | \( 1 + (1.98e3 + 822. i)T + (1.37e7 + 1.37e7i)T^{2} \) |
| 13 | \( 1 + (-3.51e3 - 8.48e3i)T + (-4.43e7 + 4.43e7i)T^{2} \) |
| 17 | \( 1 + 6.87e3iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (-5.40e3 - 1.30e4i)T + (-6.32e8 + 6.32e8i)T^{2} \) |
| 23 | \( 1 + (7.91e3 - 7.91e3i)T - 3.40e9iT^{2} \) |
| 29 | \( 1 + (-5.10e4 + 2.11e4i)T + (1.21e10 - 1.21e10i)T^{2} \) |
| 31 | \( 1 - 1.21e3T + 2.75e10T^{2} \) |
| 37 | \( 1 + (4.95e4 - 1.19e5i)T + (-6.71e10 - 6.71e10i)T^{2} \) |
| 41 | \( 1 + (-2.52e5 + 2.52e5i)T - 1.94e11iT^{2} \) |
| 43 | \( 1 + (7.52e5 + 3.11e5i)T + (1.92e11 + 1.92e11i)T^{2} \) |
| 47 | \( 1 - 9.56e4iT - 5.06e11T^{2} \) |
| 53 | \( 1 + (-4.14e5 - 1.71e5i)T + (8.30e11 + 8.30e11i)T^{2} \) |
| 59 | \( 1 + (7.26e5 - 1.75e6i)T + (-1.75e12 - 1.75e12i)T^{2} \) |
| 61 | \( 1 + (2.37e6 - 9.81e5i)T + (2.22e12 - 2.22e12i)T^{2} \) |
| 67 | \( 1 + (-1.34e6 + 5.58e5i)T + (4.28e12 - 4.28e12i)T^{2} \) |
| 71 | \( 1 + (-1.24e5 - 1.24e5i)T + 9.09e12iT^{2} \) |
| 73 | \( 1 + (2.10e6 - 2.10e6i)T - 1.10e13iT^{2} \) |
| 79 | \( 1 - 6.34e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (1.62e6 + 3.91e6i)T + (-1.91e13 + 1.91e13i)T^{2} \) |
| 89 | \( 1 + (-7.06e5 - 7.06e5i)T + 4.42e13iT^{2} \) |
| 97 | \( 1 + 2.89e6T + 8.07e13T^{2} \) |
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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
−15.61095833355452190392461484632, −14.14912698445986392824469418754, −13.60836855182808640047032628255, −11.76999255909435012535372334745, −10.81134692639154014449628579967, −10.12232565550653249504543288809, −7.24026953225737219562772257298, −6.26599286978556134925193260776, −3.96552496868958406631055031538, −2.89493128097124744111476670538,
0.094040797656938116374260938964, 3.26832691532917939093226906151, 5.12918767484858428471297499081, 6.11257833697384057167504541079, 8.182265766328438545848130962287, 9.104166906618173043046734577700, 11.75384704065896041680272481482, 12.58712323947056688899970005109, 13.06656874957393121705528433556, 15.23929727321448555250685086338
