| L(s) = 1 | − 2.56·3-s − 5-s + 7-s + 3.56·9-s + 2.56·11-s − 4.56·13-s + 2.56·15-s − 4.56·17-s + 1.12·19-s − 2.56·21-s + 5.12·23-s + 25-s − 1.43·27-s + 5.68·29-s − 6.56·33-s − 35-s − 6·37-s + 11.6·39-s − 3.12·41-s + 9.12·43-s − 3.56·45-s − 3.68·47-s + 49-s + 11.6·51-s − 3.12·53-s − 2.56·55-s − 2.87·57-s + ⋯ |
| L(s) = 1 | − 1.47·3-s − 0.447·5-s + 0.377·7-s + 1.18·9-s + 0.772·11-s − 1.26·13-s + 0.661·15-s − 1.10·17-s + 0.257·19-s − 0.558·21-s + 1.06·23-s + 0.200·25-s − 0.276·27-s + 1.05·29-s − 1.14·33-s − 0.169·35-s − 0.986·37-s + 1.87·39-s − 0.487·41-s + 1.39·43-s − 0.530·45-s − 0.537·47-s + 0.142·49-s + 1.63·51-s − 0.428·53-s − 0.345·55-s − 0.381·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 11 | \( 1 - 2.56T + 11T^{2} \) |
| 13 | \( 1 + 4.56T + 13T^{2} \) |
| 17 | \( 1 + 4.56T + 17T^{2} \) |
| 19 | \( 1 - 1.12T + 19T^{2} \) |
| 23 | \( 1 - 5.12T + 23T^{2} \) |
| 29 | \( 1 - 5.68T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 3.12T + 41T^{2} \) |
| 43 | \( 1 - 9.12T + 43T^{2} \) |
| 47 | \( 1 + 3.68T + 47T^{2} \) |
| 53 | \( 1 + 3.12T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 9.36T + 61T^{2} \) |
| 67 | \( 1 + 6.24T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 4.24T + 73T^{2} \) |
| 79 | \( 1 - 6.56T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 7.12T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−8.726926672716542490616163964383, −7.67373879751037705536044778130, −6.85247376633402177542848197270, −6.46813871385010048779160990073, −5.28244445212434510652273163418, −4.83628751506322395391220138946, −4.04932691335292093910028173888, −2.66344714410800385053173730332, −1.23981218392369978209971089540, 0,
1.23981218392369978209971089540, 2.66344714410800385053173730332, 4.04932691335292093910028173888, 4.83628751506322395391220138946, 5.28244445212434510652273163418, 6.46813871385010048779160990073, 6.85247376633402177542848197270, 7.67373879751037705536044778130, 8.726926672716542490616163964383
