L(s) = 1 | − 6i·5-s + 21.1·7-s − 42.3i·11-s + 20i·13-s − 8·17-s + 84.6i·19-s + 169.·23-s + 89·25-s − 46i·29-s − 21.1·31-s − 126. i·35-s + 164i·37-s + 312·41-s − 423. i·43-s − 169.·47-s + ⋯ |
L(s) = 1 | − 0.536i·5-s + 1.14·7-s − 1.16i·11-s + 0.426i·13-s − 0.114·17-s + 1.02i·19-s + 1.53·23-s + 0.711·25-s − 0.294i·29-s − 0.122·31-s − 0.613i·35-s + 0.728i·37-s + 1.18·41-s − 1.50i·43-s − 0.525·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.567375465\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.567375465\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 6iT - 125T^{2} \) |
| 7 | \( 1 - 21.1T + 343T^{2} \) |
| 11 | \( 1 + 42.3iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 20iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 84.6iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 169.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 46iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 21.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 164iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 312T + 6.89e4T^{2} \) |
| 43 | \( 1 + 423. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 169.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 266iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 253. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 132iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 507. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 677.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 246T + 3.89e5T^{2} \) |
| 79 | \( 1 - 232.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 973. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.39e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 302T + 9.12e5T^{2} \) |
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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
−9.032959481127559728152602678245, −8.579720491216612238277776743613, −7.84444744088947442729253045814, −6.86247919410912032856713318743, −5.77377340618477275039220994881, −5.05403635175349150163910024257, −4.18917185355927322950393102587, −3.04778070124184201568645686863, −1.68938816589699488749758997620, −0.76254989645313024606813594334,
1.01512033486156650575088197395, 2.20643780495507675980572054619, 3.15862577324839819390455785523, 4.63306543984598574957539009257, 4.94998026026571007814556191358, 6.26186390420236299262986898798, 7.23686578783479364261199594819, 7.65418099098983203744304805846, 8.788701521465357771514201068547, 9.428835738369168115366466927328