| L(s) = 1 | + i·3-s − i·5-s − 3·7-s + 2·9-s + i·11-s − 6i·13-s + 15-s − 3·17-s + 7i·19-s − 3i·21-s − 25-s + 5i·27-s + 3i·29-s + 9·31-s − 33-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.447i·5-s − 1.13·7-s + 0.666·9-s + 0.301i·11-s − 1.66i·13-s + 0.258·15-s − 0.727·17-s + 1.60i·19-s − 0.654i·21-s − 0.200·25-s + 0.962i·27-s + 0.557i·29-s + 1.61·31-s − 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.366562810\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.366562810\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 11 | \( 1 - iT \) |
| good | 3 | \( 1 - iT - 3T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 7iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 3iT - 29T^{2} \) |
| 31 | \( 1 - 9T + 31T^{2} \) |
| 37 | \( 1 - 3iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 10iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 9iT - 53T^{2} \) |
| 59 | \( 1 + 12iT - 59T^{2} \) |
| 61 | \( 1 + 15iT - 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 - 15T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−8.415317969902969917082414696435, −7.916981409507803726665256838727, −6.89754845049490034980211163068, −6.24660613532683398149998267182, −5.36368849850428716155194170398, −4.68645619669585650864443441908, −3.68296912556040364652940384020, −3.20118439411444718005489213642, −1.86820043939579929487053837639, −0.49220263621918546149351696304,
0.968452618893873948892995890884, 2.28244991752619073073758111324, 2.90448748705585543412542777486, 4.15969480470237064037497725411, 4.59586677401671401201174581791, 6.10977367590028399572312144725, 6.51500374895126969357557026780, 7.01711699713776818733449906117, 7.69030964902408387165170089286, 8.884650437454509526240837473480
