| L(s) = 1 | + 3-s + 2·4-s − 3i·7-s + 9-s − 3i·11-s + 2·12-s + (−2 − 3i)13-s + 4·16-s − 3·17-s − 3i·21-s + 3·23-s + 27-s − 6i·28-s − 6·29-s − 6i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 4-s − 1.13i·7-s + 0.333·9-s − 0.904i·11-s + 0.577·12-s + (−0.554 − 0.832i)13-s + 16-s − 0.727·17-s − 0.654i·21-s + 0.625·23-s + 0.192·27-s − 1.13i·28-s − 1.11·29-s − 1.07i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.07286 - 1.10936i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.07286 - 1.10936i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (2 + 3i)T \) |
| good | 2 | \( 1 - 2T^{2} \) |
| 7 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 + 3iT - 11T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 6iT - 31T^{2} \) |
| 37 | \( 1 - 9iT - 37T^{2} \) |
| 41 | \( 1 - 3iT - 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 - 12iT - 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 9iT - 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 - T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 - 15iT - 89T^{2} \) |
| 97 | \( 1 - 9iT - 97T^{2} \) |
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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
−9.983133119649501121485329400809, −9.062140486490919815802981401847, −7.892480404282757024996330305589, −7.53542134176631198358354871038, −6.60410925044539942272181830883, −5.71115413057068205718745805993, −4.38935995798510647778251278164, −3.34249178551493308952259467660, −2.50463904316358377716741903455, −1.02031043262301699410085471867,
1.98986558693205345866975887997, 2.37996971644120842624565130602, 3.69193104071670533041584123181, 4.93994687541428352581621825825, 5.90701223857490585696535836757, 7.03093684242178944687350142204, 7.34594096643427722638850360868, 8.651257462237378221770982044723, 9.181199570012231025496148979138, 10.06065802446174501461455096350
