| L(s) = 1 | − 3.11i·3-s − i·5-s + 1.38·7-s − 6.71·9-s − 5.13·11-s + 5.71·13-s − 3.11·15-s − 4.71i·17-s − 8.44·19-s − 4.31i·21-s + (−0.287 + 4.78i)23-s − 25-s + 11.5i·27-s − 4.10·29-s + 6.74i·31-s + ⋯ |
| L(s) = 1 | − 1.79i·3-s − 0.447i·5-s + 0.523·7-s − 2.23·9-s − 1.54·11-s + 1.58·13-s − 0.804·15-s − 1.14i·17-s − 1.93·19-s − 0.941i·21-s + (−0.0598 + 0.998i)23-s − 0.200·25-s + 2.22i·27-s − 0.762·29-s + 1.21i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6371266722\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6371266722\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (0.287 - 4.78i)T \) |
| good | 3 | \( 1 + 3.11iT - 3T^{2} \) |
| 7 | \( 1 - 1.38T + 7T^{2} \) |
| 11 | \( 1 + 5.13T + 11T^{2} \) |
| 13 | \( 1 - 5.71T + 13T^{2} \) |
| 17 | \( 1 + 4.71iT - 17T^{2} \) |
| 19 | \( 1 + 8.44T + 19T^{2} \) |
| 29 | \( 1 + 4.10T + 29T^{2} \) |
| 31 | \( 1 - 6.74iT - 31T^{2} \) |
| 37 | \( 1 + 1.81iT - 37T^{2} \) |
| 41 | \( 1 - 4.20T + 41T^{2} \) |
| 43 | \( 1 + 2.76T + 43T^{2} \) |
| 47 | \( 1 + 1.44iT - 47T^{2} \) |
| 53 | \( 1 + 9.60iT - 53T^{2} \) |
| 59 | \( 1 + 13.9iT - 59T^{2} \) |
| 61 | \( 1 - 5.90iT - 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 + 3.94iT - 71T^{2} \) |
| 73 | \( 1 + 3.12T + 73T^{2} \) |
| 79 | \( 1 + 9.45T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 - 13.4iT - 89T^{2} \) |
| 97 | \( 1 - 4.26iT - 97T^{2} \) |
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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
−8.390275922436043996389714811825, −7.991892857488270256528009098222, −7.23490510669306412283490613814, −6.39478328781487403937799829738, −5.64836092762130962403404411287, −4.87200480122076433308737772578, −3.38987156033193923660741786651, −2.25570047114611542828772400996, −1.49642163045498267257201289864, −0.22197047681885647524057009489,
2.19208224118713960449067571079, 3.22280917829305931924619417762, 4.18162948637782822785880487014, 4.58660581216957190366418749229, 5.88886084369113895803251805550, 6.07611889604319801297382426893, 7.74253791022267588499824482853, 8.493384964489617213133227679029, 8.858296305273361935265550515193, 10.06212622620167216992096081148
