| L(s) = 1 | − 3·3-s − 2i·5-s − i·7-s + 6·9-s + 5i·11-s + 6i·15-s − 3·17-s − 3i·19-s + 3i·21-s − 23-s + 25-s − 9·27-s − 29-s − 8i·31-s − 15i·33-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.894i·5-s − 0.377i·7-s + 2·9-s + 1.50i·11-s + 1.54i·15-s − 0.727·17-s − 0.688i·19-s + 0.654i·21-s − 0.208·23-s + 0.200·25-s − 1.73·27-s − 0.185·29-s − 1.43i·31-s − 2.61i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 3T + 3T^{2} \) |
| 5 | \( 1 + 2iT - 5T^{2} \) |
| 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 - 5iT - 11T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + 3iT - 19T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 + 8iT - 31T^{2} \) |
| 37 | \( 1 - 3iT - 37T^{2} \) |
| 41 | \( 1 + 3iT - 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 5iT - 59T^{2} \) |
| 61 | \( 1 + 5T + 61T^{2} \) |
| 67 | \( 1 - 7iT - 67T^{2} \) |
| 71 | \( 1 + 11iT - 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 9iT - 89T^{2} \) |
| 97 | \( 1 - iT - 97T^{2} \) |
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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
−8.355990101233446421508518074115, −7.29214797060680267001806457103, −6.86778345838691960314820289972, −5.99654022553475526064843358815, −5.16446757882146182763342433598, −4.60278737951000839790896387538, −4.09098842476873537029941688833, −2.19182737520202197889834736252, −1.08774812531711461237423951319, 0,
1.33614427190717235614202331868, 2.79505117336957982925488444193, 3.76828823816965994463499085942, 4.82658152974145223715346668608, 5.62741914629919303579721229697, 6.20262203950852294553282350452, 6.67276623521362910482932852069, 7.54613040725711209323396306153, 8.529452310642045173861355769014
