| L(s) = 1 | − 3-s + 1.56·5-s − 5.12·7-s + 9-s − 2.43·11-s − 3.56·13-s − 1.56·15-s − 17-s + 4.68·19-s + 5.12·21-s − 7.56·23-s − 2.56·25-s − 27-s − 7.12·29-s + 8.24·31-s + 2.43·33-s − 8·35-s − 4·37-s + 3.56·39-s − 2.68·41-s − 4.68·43-s + 1.56·45-s − 0.876·47-s + 19.2·49-s + 51-s + 6·53-s − 3.80·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.698·5-s − 1.93·7-s + 0.333·9-s − 0.735·11-s − 0.987·13-s − 0.403·15-s − 0.242·17-s + 1.07·19-s + 1.11·21-s − 1.57·23-s − 0.512·25-s − 0.192·27-s − 1.32·29-s + 1.48·31-s + 0.424·33-s − 1.35·35-s − 0.657·37-s + 0.570·39-s − 0.419·41-s − 0.714·43-s + 0.232·45-s − 0.127·47-s + 2.74·49-s + 0.140·51-s + 0.824·53-s − 0.513·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 1.56T + 5T^{2} \) |
| 7 | \( 1 + 5.12T + 7T^{2} \) |
| 11 | \( 1 + 2.43T + 11T^{2} \) |
| 13 | \( 1 + 3.56T + 13T^{2} \) |
| 19 | \( 1 - 4.68T + 19T^{2} \) |
| 23 | \( 1 + 7.56T + 23T^{2} \) |
| 29 | \( 1 + 7.12T + 29T^{2} \) |
| 31 | \( 1 - 8.24T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 2.68T + 41T^{2} \) |
| 43 | \( 1 + 4.68T + 43T^{2} \) |
| 47 | \( 1 + 0.876T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 8.24T + 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 - 7.12T + 83T^{2} \) |
| 89 | \( 1 - 9.12T + 89T^{2} \) |
| 97 | \( 1 - 1.12T + 97T^{2} \) |
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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
−10.51702695396385956953822481930, −9.777417715972448284387164144686, −9.506182534472836331713918150557, −7.85146766862393415742459487371, −6.79812835845347119190101470810, −6.02854530138363561052310094010, −5.18783724358565458895793328434, −3.62593992355196624722673370039, −2.37575072165824875931780526192, 0,
2.37575072165824875931780526192, 3.62593992355196624722673370039, 5.18783724358565458895793328434, 6.02854530138363561052310094010, 6.79812835845347119190101470810, 7.85146766862393415742459487371, 9.506182534472836331713918150557, 9.777417715972448284387164144686, 10.51702695396385956953822481930
