L(s) = 1 | − 0.352i·2-s − 3i·3-s + 7.87·4-s + (10.7 − 3.10i)5-s − 1.05·6-s − 19.8i·7-s − 5.59i·8-s − 9·9-s + (−1.09 − 3.78i)10-s + 11·11-s − 23.6i·12-s + 25.5i·13-s − 6.99·14-s + (−9.31 − 32.2i)15-s + 61.0·16-s + 77.3i·17-s + ⋯ |
L(s) = 1 | − 0.124i·2-s − 0.577i·3-s + 0.984·4-s + (0.960 − 0.277i)5-s − 0.0719·6-s − 1.07i·7-s − 0.247i·8-s − 0.333·9-s + (−0.0346 − 0.119i)10-s + 0.301·11-s − 0.568i·12-s + 0.544i·13-s − 0.133·14-s + (−0.160 − 0.554i)15-s + 0.953·16-s + 1.10i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.92734 - 1.44894i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.92734 - 1.44894i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3iT \) |
| 5 | \( 1 + (-10.7 + 3.10i)T \) |
| 11 | \( 1 - 11T \) |
good | 2 | \( 1 + 0.352iT - 8T^{2} \) |
| 7 | \( 1 + 19.8iT - 343T^{2} \) |
| 13 | \( 1 - 25.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 77.3iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 96.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 173. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 189.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 275.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 269. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 306.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 477. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 257. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 495. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 102.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 585.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 474. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 453.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 655. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 482.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 523. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 750.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 572. iT - 9.12e5T^{2} \) |
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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
−12.46895784047571017888444412285, −10.88149098266790050646835354020, −10.53404571816439166023702116577, −9.110788718652964293003723046735, −7.84333013136191731380881240337, −6.64407836381918711988305257438, −6.12355410611255905839643032033, −4.23851852517338934134318109786, −2.39136127690927239848987522382, −1.23009925026891333598775666200,
2.03121693342001791795399043490, 3.13612091066459653405592651637, 5.26220496222884729149507600554, 6.00926730155861420511083022653, 7.14459841372930831617569674178, 8.639021781611077911417032760772, 9.605181654552690471792690419554, 10.56557423638720895854811625222, 11.48605317476597456245891168913, 12.39882237805231632860104955745