L(s) = 1 | − 3.23·2-s + (3.88 + 3.45i)3-s + 2.45·4-s + (−8.12 − 7.68i)5-s + (−12.5 − 11.1i)6-s + (17.9 + 4.72i)7-s + 17.9·8-s + (3.15 + 26.8i)9-s + (26.2 + 24.8i)10-s − 0.605i·11-s + (9.52 + 8.47i)12-s − 12.8·13-s + (−57.8 − 15.2i)14-s + (−5.01 − 57.8i)15-s − 77.6·16-s + 117. i·17-s + ⋯ |
L(s) = 1 | − 1.14·2-s + (0.747 + 0.664i)3-s + 0.306·4-s + (−0.726 − 0.687i)5-s + (−0.854 − 0.759i)6-s + (0.966 + 0.255i)7-s + 0.792·8-s + (0.116 + 0.993i)9-s + (0.830 + 0.785i)10-s − 0.0165i·11-s + (0.229 + 0.203i)12-s − 0.273·13-s + (−1.10 − 0.291i)14-s + (−0.0863 − 0.996i)15-s − 1.21·16-s + 1.68i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.170 - 0.985i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.170 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.712644 + 0.599702i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.712644 + 0.599702i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-3.88 - 3.45i)T \) |
| 5 | \( 1 + (8.12 + 7.68i)T \) |
| 7 | \( 1 + (-17.9 - 4.72i)T \) |
good | 2 | \( 1 + 3.23T + 8T^{2} \) |
| 11 | \( 1 + 0.605iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 12.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 117. iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 98.5iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 136.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 77.5iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 131. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 260. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 58.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 519. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 104. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 550.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 498.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 172. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 622. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 151. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 242.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 94.6T + 4.93e5T^{2} \) |
| 83 | \( 1 - 779. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.00e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.12e3T + 9.12e5T^{2} \) |
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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
−13.63336877624294921889471936255, −12.33933332423748393255166896479, −10.98516429460492965665563197571, −10.15966764424526283092564692389, −8.854680106020198494328325270345, −8.381347336875224782756666656563, −7.57851591456059797752606098700, −5.07756511186734451138296370299, −3.95178755937964511371500404314, −1.60462095299975087616529264951,
0.77201854136246420283738917119, 2.65654426306426204374418844141, 4.57103736219579380095754899562, 7.16774556346189318633255097069, 7.45357368599713357581959639032, 8.624093332832935457136689878959, 9.494866857627333449752739792062, 10.93636102287271712191908392101, 11.64248454252332810477211983460, 13.20288709073836664540213880092