| L(s) = 1 | − i·3-s + (1.32 − 1.80i)5-s − 9-s + 4.64·11-s + i·13-s + (−1.80 − 1.32i)15-s − 4.24i·17-s − 6.24·19-s − 2.24i·23-s + (−1.51 − 4.76i)25-s + i·27-s + 9.21·29-s − 9.28·31-s − 4.64i·33-s − 7.28i·37-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (0.590 − 0.807i)5-s − 0.333·9-s + 1.39·11-s + 0.277i·13-s + (−0.466 − 0.340i)15-s − 1.03i·17-s − 1.43·19-s − 0.469i·23-s + (−0.303 − 0.952i)25-s + 0.192i·27-s + 1.71·29-s − 1.66·31-s − 0.807i·33-s − 1.19i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.590 + 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.590 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.857768504\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.857768504\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-1.32 + 1.80i)T \) |
| 13 | \( 1 - iT \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 4.64T + 11T^{2} \) |
| 17 | \( 1 + 4.24iT - 17T^{2} \) |
| 19 | \( 1 + 6.24T + 19T^{2} \) |
| 23 | \( 1 + 2.24iT - 23T^{2} \) |
| 29 | \( 1 - 9.21T + 29T^{2} \) |
| 31 | \( 1 + 9.28T + 31T^{2} \) |
| 37 | \( 1 + 7.28iT - 37T^{2} \) |
| 41 | \( 1 + 5.67T + 41T^{2} \) |
| 43 | \( 1 - 4.24iT - 43T^{2} \) |
| 47 | \( 1 - 2.88iT - 47T^{2} \) |
| 53 | \( 1 + 9.21iT - 53T^{2} \) |
| 59 | \( 1 - 5.92T + 59T^{2} \) |
| 61 | \( 1 - 0.969T + 61T^{2} \) |
| 67 | \( 1 + 1.93iT - 67T^{2} \) |
| 71 | \( 1 + 5.60T + 71T^{2} \) |
| 73 | \( 1 + 12.5iT - 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 - 3.67iT - 83T^{2} \) |
| 89 | \( 1 - 9.67T + 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−8.702160125020813854334287325133, −7.69070463292639961085106219391, −6.66900087466562694618685242862, −6.41164177992001988049353779881, −5.38808955063404529118705020526, −4.58876805687438348338886408060, −3.77789327831133070368561813568, −2.43635022999084056155742735816, −1.65211422586707982345360402373, −0.57406262764946145722890282107,
1.45748957006103370277085726975, 2.45703590356028158849151447077, 3.56479881321858100176329149335, 4.07280584953882041810949398383, 5.16021904409751230792705615748, 6.10597419909817265755560644654, 6.51155025867433857082019268527, 7.31305966201475698158461744232, 8.511780823662353010529048227511, 8.866125595092648902266858081011
