| L(s) = 1 | + (8 − 13.8i)2-s + (9.29 − 16.0i)3-s + (−127. − 221. i)4-s + (830. − 1.43e3i)5-s + (−148. − 257. i)6-s + 1.05e4·7-s − 4.09e3·8-s + (9.66e3 + 1.67e4i)9-s + (−1.32e4 − 2.30e4i)10-s − 1.25e3·11-s − 4.75e3·12-s + (−2.59e4 − 4.49e4i)13-s + (8.41e4 − 1.45e5i)14-s + (−1.54e4 − 2.67e4i)15-s + (−3.27e4 + 5.67e4i)16-s + (6.36e4 − 1.10e5i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (0.0662 − 0.114i)3-s + (−0.249 − 0.433i)4-s + (0.594 − 1.02i)5-s + (−0.0468 − 0.0811i)6-s + 1.65·7-s − 0.353·8-s + (0.491 + 0.850i)9-s + (−0.420 − 0.728i)10-s − 0.0258·11-s − 0.0662·12-s + (−0.252 − 0.436i)13-s + (0.585 − 1.01i)14-s + (−0.0787 − 0.136i)15-s + (−0.125 + 0.216i)16-s + (0.184 − 0.320i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.211 + 0.977i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.86729 - 2.31533i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.86729 - 2.31533i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-8 + 13.8i)T \) |
| 19 | \( 1 + (-2.34e5 + 5.17e5i)T \) |
| good | 3 | \( 1 + (-9.29 + 16.0i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-830. + 1.43e3i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 7 | \( 1 - 1.05e4T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.25e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + (2.59e4 + 4.49e4i)T + (-5.30e9 + 9.18e9i)T^{2} \) |
| 17 | \( 1 + (-6.36e4 + 1.10e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 23 | \( 1 + (5.63e5 + 9.76e5i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + (2.54e5 + 4.41e5i)T + (-7.25e12 + 1.25e13i)T^{2} \) |
| 31 | \( 1 + 5.84e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 4.24e5T + 1.29e14T^{2} \) |
| 41 | \( 1 + (2.46e5 - 4.27e5i)T + (-1.63e14 - 2.83e14i)T^{2} \) |
| 43 | \( 1 + (9.44e6 - 1.63e7i)T + (-2.51e14 - 4.35e14i)T^{2} \) |
| 47 | \( 1 + (-2.97e6 - 5.14e6i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (7.52e6 + 1.30e7i)T + (-1.64e15 + 2.85e15i)T^{2} \) |
| 59 | \( 1 + (5.37e7 - 9.31e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-8.28e7 - 1.43e8i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-9.61e7 - 1.66e8i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + (4.00e6 - 6.93e6i)T + (-2.29e16 - 3.97e16i)T^{2} \) |
| 73 | \( 1 + (-1.81e8 + 3.13e8i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-2.90e8 + 5.02e8i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 - 3.99e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (5.32e8 + 9.21e8i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + (1.86e8 - 3.23e8i)T + (-3.80e17 - 6.58e17i)T^{2} \) |
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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
−13.73568140216287165770747503676, −12.84709691788718157531607976417, −11.58255900638057193312496029587, −10.43426385694038545969211139438, −8.964303158879863629451130756232, −7.71654291306224930152048125993, −5.30156512053900290322794205950, −4.63623173889208259164771930915, −2.16330331409301756283520941983, −1.09096860594422330507313248152,
1.78833856981667030523428377321, 3.77919399962238405173325962136, 5.38939850931832378237917351993, 6.79836741623432161662065281809, 8.006675280519268904114910854127, 9.632779204167714431613872695902, 11.01974261598037259713711477914, 12.26838017038523141007270160379, 14.02226656633643581061617856259, 14.51765871132677312623623155554
