| L(s) = 1 | − 4.76·3-s + 15.5·7-s − 4.30·9-s − 74.3·21-s − 207.·23-s + 149.·27-s + 306·29-s + 460.·41-s + 30.9·43-s − 643.·47-s − 99.7·49-s + 40.2·61-s − 67.1·63-s − 1.09e3·67-s + 990.·69-s − 594.·81-s + 1.14e3·83-s − 1.45e3·87-s − 1.38e3·89-s − 378·101-s − 1.98e3·103-s − 1.77e3·107-s − 1.97e3·109-s + ⋯ |
| L(s) = 1 | − 0.916·3-s + 0.842·7-s − 0.159·9-s − 0.772·21-s − 1.88·23-s + 1.06·27-s + 1.95·29-s + 1.75·41-s + 0.109·43-s − 1.99·47-s − 0.290·49-s + 0.0844·61-s − 0.134·63-s − 1.99·67-s + 1.72·69-s − 0.815·81-s + 1.51·83-s − 1.79·87-s − 1.65·89-s − 0.372·101-s − 1.89·103-s − 1.59·107-s − 1.73·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 4.76T + 27T^{2} \) |
| 7 | \( 1 - 15.5T + 343T^{2} \) |
| 11 | \( 1 + 1.33e3T^{2} \) |
| 13 | \( 1 + 2.19e3T^{2} \) |
| 17 | \( 1 + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.85e3T^{2} \) |
| 23 | \( 1 + 207.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 306T + 2.43e4T^{2} \) |
| 31 | \( 1 + 2.97e4T^{2} \) |
| 37 | \( 1 + 5.06e4T^{2} \) |
| 41 | \( 1 - 460.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 30.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 643.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 1.48e5T^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 - 40.2T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.09e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 + 3.89e5T^{2} \) |
| 79 | \( 1 + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.14e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.38e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 9.12e5T^{2} \) |
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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
−9.598496039081116197097180137503, −8.398494632412803378676961684892, −7.88140100415785314858468957853, −6.60320901499457661957472492358, −5.92468466338773291617714966358, −4.99958247131198918555662384359, −4.20623904874380482717792496637, −2.69985357930334609001843806263, −1.34296313056487185585076265782, 0,
1.34296313056487185585076265782, 2.69985357930334609001843806263, 4.20623904874380482717792496637, 4.99958247131198918555662384359, 5.92468466338773291617714966358, 6.60320901499457661957472492358, 7.88140100415785314858468957853, 8.398494632412803378676961684892, 9.598496039081116197097180137503
