| L(s) = 1 | − 0.859i·2-s + i·3-s + 1.26·4-s + i·5-s + 0.859·6-s + (−2.42 − 1.05i)7-s − 2.80i·8-s − 9-s + 0.859·10-s + (−3.22 − 0.765i)11-s + 1.26i·12-s + 3.49·13-s + (−0.904 + 2.08i)14-s − 15-s + 0.110·16-s + 5.22·17-s + ⋯ |
| L(s) = 1 | − 0.608i·2-s + 0.577i·3-s + 0.630·4-s + 0.447i·5-s + 0.351·6-s + (−0.917 − 0.397i)7-s − 0.991i·8-s − 0.333·9-s + 0.271·10-s + (−0.972 − 0.230i)11-s + 0.363i·12-s + 0.968·13-s + (−0.241 + 0.557i)14-s − 0.258·15-s + 0.0275·16-s + 1.26·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 + 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.801 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.845781664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.845781664\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (2.42 + 1.05i)T \) |
| 11 | \( 1 + (3.22 + 0.765i)T \) |
| good | 2 | \( 1 + 0.859iT - 2T^{2} \) |
| 13 | \( 1 - 3.49T + 13T^{2} \) |
| 17 | \( 1 - 5.22T + 17T^{2} \) |
| 19 | \( 1 - 4.83T + 19T^{2} \) |
| 23 | \( 1 - 3.63T + 23T^{2} \) |
| 29 | \( 1 + 6.88iT - 29T^{2} \) |
| 31 | \( 1 + 1.77iT - 31T^{2} \) |
| 37 | \( 1 - 9.57T + 37T^{2} \) |
| 41 | \( 1 + 2.91T + 41T^{2} \) |
| 43 | \( 1 - 4.10iT - 43T^{2} \) |
| 47 | \( 1 - 6.68iT - 47T^{2} \) |
| 53 | \( 1 - 7.05T + 53T^{2} \) |
| 59 | \( 1 - 4.22iT - 59T^{2} \) |
| 61 | \( 1 - 3.39T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 + 7.34T + 73T^{2} \) |
| 79 | \( 1 + 6.63iT - 79T^{2} \) |
| 83 | \( 1 + 7.78T + 83T^{2} \) |
| 89 | \( 1 + 13.2iT - 89T^{2} \) |
| 97 | \( 1 + 12.8iT - 97T^{2} \) |
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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
−10.05953999266562718043550711628, −9.270269574255503907179629189649, −7.930160477653063043357889677233, −7.31466832663438611513397478250, −6.20401161731599141750069250901, −5.64512987116812798200404150625, −4.13854400890804090542213398675, −3.20696400727504540502539602144, −2.76679800132636399071215434700, −0.956782024917900070070633092183,
1.20523750108851127264743960995, 2.63379320405771352961265051576, 3.43999592477476780226318961436, 5.33995925160888369124873192187, 5.58241060661327628494125126412, 6.65867666733679388382637708567, 7.30462082783669894533847221016, 8.097530105682814879153490632048, 8.812505478441427467440405776834, 9.824520214692872857624505878851
