| L(s) = 1 | + (46.7 + 1.97i)3-s + 125i·5-s − 710. i·7-s + (2.17e3 + 184. i)9-s + 2.11e3·11-s − 1.38e4·13-s + (−246. + 5.84e3i)15-s + 2.39e4i·17-s + 2.80e4i·19-s + (1.40e3 − 3.32e4i)21-s + 4.68e3·23-s − 1.56e4·25-s + (1.01e5 + 1.29e4i)27-s + 2.04e5i·29-s − 2.93e5i·31-s + ⋯ |
| L(s) = 1 | + (0.999 + 0.0422i)3-s + 0.447i·5-s − 0.783i·7-s + (0.996 + 0.0843i)9-s + 0.478·11-s − 1.75·13-s + (−0.0188 + 0.446i)15-s + 1.18i·17-s + 0.937i·19-s + (0.0330 − 0.782i)21-s + 0.0803·23-s − 0.199·25-s + (0.991 + 0.126i)27-s + 1.55i·29-s − 1.76i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0422 - 0.999i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.0422 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.475459951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.475459951\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-46.7 - 1.97i)T \) |
| 5 | \( 1 - 125iT \) |
| good | 7 | \( 1 + 710. iT - 8.23e5T^{2} \) |
| 11 | \( 1 - 2.11e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.38e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.39e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 2.80e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 - 4.68e3T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.04e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 2.93e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 - 1.33e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.93e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 - 8.24e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 1.08e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.96e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 2.40e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.95e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.22e5iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 1.63e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.96e4T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.73e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 - 6.45e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.16e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 + 3.73e5T + 8.07e13T^{2} \) |
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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
−10.86583961246154842286015045128, −10.01551562241565250527643701013, −9.326868050498195763371628094765, −7.955144657258949205936519928085, −7.41392526688764706306006315429, −6.29124476299503315317838534783, −4.59275113837848545194184060924, −3.68633525157998530095037753849, −2.52620071081362118251234267322, −1.32630544835107594578023579005,
0.49257028280582247428447425915, 2.10502995843730768355131255957, 2.86449643246279160492718948842, 4.38276116977283812426064538822, 5.30050343878130387510897865629, 6.92661483237151027029569815301, 7.67931746579278852665426225122, 9.066192318359656985389630765166, 9.207665972381308449238227503482, 10.41013870251154974245769881294
