L(s) = 1 | + (−1 + i)2-s − i·3-s − 2i·4-s − 3.74i·5-s + (1 + i)6-s − 3.74·7-s + (2 + 2i)8-s + 2·9-s + (3.74 + 3.74i)10-s + 3.74·11-s − 2·12-s − 13-s + (3.74 − 3.74i)14-s − 3.74·15-s − 4·16-s + 3.74i·17-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)2-s − 0.577i·3-s − i·4-s − 1.67i·5-s + (0.408 + 0.408i)6-s − 1.41·7-s + (0.707 + 0.707i)8-s + 0.666·9-s + (1.18 + 1.18i)10-s + 1.12·11-s − 0.577·12-s − 0.277·13-s + (0.999 − 0.999i)14-s − 0.966·15-s − 16-s + 0.907i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.603489 - 0.289648i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.603489 - 0.289648i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 - i)T \) |
| 23 | \( 1 + (-3.74 + 3i)T \) |
good | 3 | \( 1 + iT - 3T^{2} \) |
| 5 | \( 1 + 3.74iT - 5T^{2} \) |
| 7 | \( 1 + 3.74T + 7T^{2} \) |
| 11 | \( 1 - 3.74T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 3.74iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 - 5iT - 31T^{2} \) |
| 37 | \( 1 - 3.74iT - 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 7.48T + 43T^{2} \) |
| 47 | \( 1 + 3iT - 47T^{2} \) |
| 53 | \( 1 - 7.48iT - 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 3.74T + 67T^{2} \) |
| 71 | \( 1 + 5iT - 71T^{2} \) |
| 73 | \( 1 + T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 + 7.48iT - 89T^{2} \) |
| 97 | \( 1 - 3.74iT - 97T^{2} \) |
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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
−13.79139756592808700644314469031, −12.81509898799787686570190640369, −12.18365734009990219137267444108, −10.20502480746488792986201362112, −9.247705733171638758499153361632, −8.498728099871941280755118620706, −7.02967586411552443769009147024, −6.09537738027108013529802472511, −4.47854774855280332112947046928, −1.17512742785760637790603110853,
2.88379729438543074857811845954, 3.87646399942521206091126630017, 6.61271282106728524039149849772, 7.26909014234119246530747526477, 9.367053502642331049008925863655, 9.825539837070753814876225330174, 10.78942035042762801911326979284, 11.76415115977466206498956713369, 13.01184896254838892039007840476, 14.20331243209216866946525903823