L(s) = 1 | − 3i·3-s + 2.27i·5-s + 7·7-s − 9·9-s + 1.28i·11-s − 84.9i·13-s + 6.81·15-s + 49.4·17-s + 2.27i·19-s − 21i·21-s + 96.7·23-s + 119.·25-s + 27i·27-s + 216. i·29-s − 108.·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 0.203i·5-s + 0.377·7-s − 0.333·9-s + 0.0353i·11-s − 1.81i·13-s + 0.117·15-s + 0.705·17-s + 0.0274i·19-s − 0.218i·21-s + 0.876·23-s + 0.958·25-s + 0.192i·27-s + 1.38i·29-s − 0.627·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.951898799\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.951898799\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| 7 | \( 1 - 7T \) |
good | 5 | \( 1 - 2.27iT - 125T^{2} \) |
| 11 | \( 1 - 1.28iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 84.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 49.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 2.27iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 96.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 216. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 108.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 92.2iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 356.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 500. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 104.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 53.9iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 395. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 76.1iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 55.9iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 1.00e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 149.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 331.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 729. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 7.31T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.76e3T + 9.12e5T^{2} \) |
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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
−8.758209490490435769854256286795, −8.206568578503625774795240943625, −7.33259093268957654013657067454, −6.73576249207838706135591983420, −5.46023954761251920167542537282, −5.14484166184404927506124941201, −3.52063015803336278246176463167, −2.84528192444267267526625367606, −1.51393825940625479568091679964, −0.49791728486335156370207241665,
1.12669189911068873763081550270, 2.30744975684723680742373371495, 3.53462449793429516166092252609, 4.45757933429880283975332568474, 5.08147540038644330605299573018, 6.16630930553001914357974114469, 7.00530775461805609071589098189, 7.948737555383040058037230657515, 8.870841266274150896256561122871, 9.369505256227247967521205754577