| L(s) = 1 | + 3.19i·3-s + i·5-s + 0.255·7-s − 7.23·9-s + 1.88i·11-s + 3.02i·13-s − 3.19·15-s − 6.87·17-s + i·19-s + 0.818i·21-s + 3.56·23-s − 25-s − 13.5i·27-s + 2.75i·29-s − 2.14·31-s + ⋯ |
| L(s) = 1 | + 1.84i·3-s + 0.447i·5-s + 0.0966·7-s − 2.41·9-s + 0.568i·11-s + 0.838i·13-s − 0.826·15-s − 1.66·17-s + 0.229i·19-s + 0.178i·21-s + 0.744·23-s − 0.200·25-s − 2.60i·27-s + 0.512i·29-s − 0.385·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.121 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.121 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7457048165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7457048165\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 19 | \( 1 - iT \) |
| good | 3 | \( 1 - 3.19iT - 3T^{2} \) |
| 7 | \( 1 - 0.255T + 7T^{2} \) |
| 11 | \( 1 - 1.88iT - 11T^{2} \) |
| 13 | \( 1 - 3.02iT - 13T^{2} \) |
| 17 | \( 1 + 6.87T + 17T^{2} \) |
| 23 | \( 1 - 3.56T + 23T^{2} \) |
| 29 | \( 1 - 2.75iT - 29T^{2} \) |
| 31 | \( 1 + 2.14T + 31T^{2} \) |
| 37 | \( 1 + 0.249iT - 37T^{2} \) |
| 41 | \( 1 - 2.59T + 41T^{2} \) |
| 43 | \( 1 - 11.2iT - 43T^{2} \) |
| 47 | \( 1 - 7.82T + 47T^{2} \) |
| 53 | \( 1 + 12.5iT - 53T^{2} \) |
| 59 | \( 1 - 7.96iT - 59T^{2} \) |
| 61 | \( 1 + 14.1iT - 61T^{2} \) |
| 67 | \( 1 + 8.77iT - 67T^{2} \) |
| 71 | \( 1 + 6.26T + 71T^{2} \) |
| 73 | \( 1 - 3.10T + 73T^{2} \) |
| 79 | \( 1 - 2.17T + 79T^{2} \) |
| 83 | \( 1 + 12.1iT - 83T^{2} \) |
| 89 | \( 1 + 3.73T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−9.263228371383446749016591795889, −8.906001640945057965891792724779, −7.943186831396057515188920970858, −6.84896336993745110190545705759, −6.20796547573367550806044132965, −5.09549399807961730816758932304, −4.58868260342533646600760387051, −3.92081101671887377059757168396, −3.04697390224170894661652791871, −2.06599398973753490781386822670,
0.24035705978500646931566292878, 1.16134132436572247031956576658, 2.24004783268532120245339873268, 2.93870487157252660849777583620, 4.25494331790543862464692461971, 5.44434244984913906921123593620, 5.92732562566655172969242673601, 6.85883520924845150020519184709, 7.29591425667789600761125917361, 8.185488412770282528093909790099
