| L(s) = 1 | − 3-s − 2.56·5-s + 3.12·7-s + 9-s − 6.56·11-s + 0.561·13-s + 2.56·15-s − 17-s − 7.68·19-s − 3.12·21-s − 3.43·23-s + 1.56·25-s − 27-s + 1.12·29-s − 8.24·31-s + 6.56·33-s − 8·35-s − 4·37-s − 0.561·39-s + 9.68·41-s + 7.68·43-s − 2.56·45-s − 9.12·47-s + 2.75·49-s + 51-s + 6·53-s + 16.8·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.14·5-s + 1.18·7-s + 0.333·9-s − 1.97·11-s + 0.155·13-s + 0.661·15-s − 0.242·17-s − 1.76·19-s − 0.681·21-s − 0.716·23-s + 0.312·25-s − 0.192·27-s + 0.208·29-s − 1.48·31-s + 1.14·33-s − 1.35·35-s − 0.657·37-s − 0.0899·39-s + 1.51·41-s + 1.17·43-s − 0.381·45-s − 1.33·47-s + 0.393·49-s + 0.140·51-s + 0.824·53-s + 2.26·55-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 | \( 1 + 2.56T + 5T^{2} \) |
| 7 | \( 1 - 3.12T + 7T^{2} \) |
| 11 | \( 1 + 6.56T + 11T^{2} \) |
| 13 | \( 1 - 0.561T + 13T^{2} \) |
| 19 | \( 1 + 7.68T + 19T^{2} \) |
| 23 | \( 1 + 3.43T + 23T^{2} \) |
| 29 | \( 1 - 1.12T + 29T^{2} \) |
| 31 | \( 1 + 8.24T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 9.68T + 41T^{2} \) |
| 43 | \( 1 - 7.68T + 43T^{2} \) |
| 47 | \( 1 + 9.12T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 + 8.24T + 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 + 1.12T + 83T^{2} \) |
| 89 | \( 1 - 0.876T + 89T^{2} \) |
| 97 | \( 1 + 7.12T + 97T^{2} \) |
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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
−10.93859900840995441485561293810, −10.29519316716338137358758958639, −8.629266634054018231760010622214, −7.957828035573267670729302141841, −7.29830978769478633052811872845, −5.82629751559987422031601475475, −4.85288075239355842650644285010, −4.00928914128834599166791049994, −2.22645922068294815811964968241, 0,
2.22645922068294815811964968241, 4.00928914128834599166791049994, 4.85288075239355842650644285010, 5.82629751559987422031601475475, 7.29830978769478633052811872845, 7.957828035573267670729302141841, 8.629266634054018231760010622214, 10.29519316716338137358758958639, 10.93859900840995441485561293810
