| L(s) = 1 | − 2·3-s + 5-s − 3·7-s + 9-s − 3·11-s − 2·15-s + 5·17-s + 2·19-s + 6·21-s + 6·23-s + 25-s + 4·27-s + 29-s + 31-s + 6·33-s − 3·35-s + 37-s − 5·41-s − 43-s + 45-s − 4·47-s + 2·49-s − 10·51-s − 9·53-s − 3·55-s − 4·57-s − 8·59-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s − 1.13·7-s + 1/3·9-s − 0.904·11-s − 0.516·15-s + 1.21·17-s + 0.458·19-s + 1.30·21-s + 1.25·23-s + 1/5·25-s + 0.769·27-s + 0.185·29-s + 0.179·31-s + 1.04·33-s − 0.507·35-s + 0.164·37-s − 0.780·41-s − 0.152·43-s + 0.149·45-s − 0.583·47-s + 2/7·49-s − 1.40·51-s − 1.23·53-s − 0.404·55-s − 0.529·57-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−8.350466132091262148465662805238, −7.43983787966264825970911146317, −6.64723802556655394861273043753, −6.05810078388367910242353322346, −5.33842654287243942726231954022, −4.83745682240847524696433124174, −3.37494820502056853872081637408, −2.78649355133592371209585121207, −1.19619062833759110393536589868, 0,
1.19619062833759110393536589868, 2.78649355133592371209585121207, 3.37494820502056853872081637408, 4.83745682240847524696433124174, 5.33842654287243942726231954022, 6.05810078388367910242353322346, 6.64723802556655394861273043753, 7.43983787966264825970911146317, 8.350466132091262148465662805238
