| L(s) = 1 | − i·2-s − i·3-s − 4-s − 1.98i·5-s − 6-s + i·8-s − 9-s − 1.98·10-s + (2.62 + 2.02i)11-s + i·12-s + 3.67·13-s − 1.98·15-s + 16-s + 5.16·17-s + i·18-s + 0.193·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.886i·5-s − 0.408·6-s + 0.353i·8-s − 0.333·9-s − 0.626·10-s + (0.791 + 0.611i)11-s + 0.288i·12-s + 1.01·13-s − 0.511·15-s + 0.250·16-s + 1.25·17-s + 0.235i·18-s + 0.0443·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.480 + 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.480 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.004490676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.004490676\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-2.62 - 2.02i)T \) |
| good | 5 | \( 1 + 1.98iT - 5T^{2} \) |
| 13 | \( 1 - 3.67T + 13T^{2} \) |
| 17 | \( 1 - 5.16T + 17T^{2} \) |
| 19 | \( 1 - 0.193T + 19T^{2} \) |
| 23 | \( 1 + 7.53T + 23T^{2} \) |
| 29 | \( 1 + 4.29iT - 29T^{2} \) |
| 31 | \( 1 - 4.34iT - 31T^{2} \) |
| 37 | \( 1 - 3.62T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 3.78iT - 43T^{2} \) |
| 47 | \( 1 - 2.69iT - 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 2.76iT - 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 - 8.97T + 67T^{2} \) |
| 71 | \( 1 + 0.605T + 71T^{2} \) |
| 73 | \( 1 + 2.49T + 73T^{2} \) |
| 79 | \( 1 + 1.40iT - 79T^{2} \) |
| 83 | \( 1 + 9.45T + 83T^{2} \) |
| 89 | \( 1 + 5.62iT - 89T^{2} \) |
| 97 | \( 1 - 9.63iT - 97T^{2} \) |
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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
−8.428151341343677354222821270304, −7.909955727287214356154604004754, −6.95995795146168056004895840707, −5.97514100291610411991155042846, −5.42513344175428096533335748734, −4.26272281214012788082131310662, −3.79012767055030975746813345966, −2.54330326414675238664409246322, −1.50168594047696000967010900607, −0.838048547284841155400920931971,
1.02753794622509274167942620209, 2.63856842448633618607081426129, 3.73417069744618361414192411350, 3.96985391125867481857135835664, 5.33677790752129814005925187634, 5.98393173690652374561691036131, 6.49080751542372780440630971028, 7.41505419873988169613465997952, 8.158030109718425521767097862835, 8.781867283479623643593953021371
