| L(s) = 1 | + (−1.58 + 1.58i)5-s − i·7-s − 3.16·11-s + 3i·13-s + 6.32i·17-s − 3·19-s + 3.16i·23-s − 5.00i·25-s − 9.48·29-s − 2·31-s + (1.58 + 1.58i)35-s − i·37-s − 3.16·41-s + 10i·43-s − 6.32i·47-s + ⋯ |
| L(s) = 1 | + (−0.707 + 0.707i)5-s − 0.377i·7-s − 0.953·11-s + 0.832i·13-s + 1.53i·17-s − 0.688·19-s + 0.659i·23-s − 1.00i·25-s − 1.76·29-s − 0.359·31-s + (0.267 + 0.267i)35-s − 0.164i·37-s − 0.493·41-s + 1.52i·43-s − 0.922i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.245100 + 0.591724i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.245100 + 0.591724i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.58 - 1.58i)T \) |
| good | 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 + 3.16T + 11T^{2} \) |
| 13 | \( 1 - 3iT - 13T^{2} \) |
| 17 | \( 1 - 6.32iT - 17T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 23 | \( 1 - 3.16iT - 23T^{2} \) |
| 29 | \( 1 + 9.48T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + iT - 37T^{2} \) |
| 41 | \( 1 + 3.16T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 + 6.32iT - 47T^{2} \) |
| 53 | \( 1 - 9.48iT - 53T^{2} \) |
| 59 | \( 1 - 6.32T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 + 11iT - 67T^{2} \) |
| 71 | \( 1 - 9.48T + 71T^{2} \) |
| 73 | \( 1 - 13iT - 73T^{2} \) |
| 79 | \( 1 - 3T + 79T^{2} \) |
| 83 | \( 1 + 15.8iT - 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 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
−10.93517731343346390211662927956, −10.56641163195596602543201659097, −9.442163148021534276849440910948, −8.295158528276832617736283795132, −7.59311716056863857875950867392, −6.72201669085319595487951365473, −5.69349141421041870368883655237, −4.29503185762542678734933904786, −3.51083706802360544452006946189, −2.04062455265025563980423584219,
0.35332328426712925442445266471, 2.40614944817258005486158744738, 3.68838905701190899944937263707, 4.97329274697948220646736498621, 5.54945835612603859208996413079, 7.06336344421532421814038642601, 7.86528528412522724079353843836, 8.653527773618160145538112893087, 9.500423991163577270739730049517, 10.58032583879914652172345147998
