| L(s) = 1 | − 3-s − 5-s + 9-s − 5·11-s + 15-s + 4·17-s + 8·19-s + 4·23-s − 4·25-s − 27-s − 5·29-s + 3·31-s + 5·33-s − 4·37-s − 2·43-s − 45-s − 6·47-s − 4·51-s − 9·53-s + 5·55-s − 8·57-s − 11·59-s + 6·61-s + 2·67-s − 4·69-s − 2·71-s − 10·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.50·11-s + 0.258·15-s + 0.970·17-s + 1.83·19-s + 0.834·23-s − 4/5·25-s − 0.192·27-s − 0.928·29-s + 0.538·31-s + 0.870·33-s − 0.657·37-s − 0.304·43-s − 0.149·45-s − 0.875·47-s − 0.560·51-s − 1.23·53-s + 0.674·55-s − 1.05·57-s − 1.43·59-s + 0.768·61-s + 0.244·67-s − 0.481·69-s − 0.237·71-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−8.447381863208563034433050526997, −7.56498833244357344695010144284, −7.39444563309779465349867586459, −6.11525795512597235717276121264, −5.32200919003813187335704029210, −4.88716868227948365972096587587, −3.58706367992406002583482936990, −2.86466238235176706687444228967, −1.37926999870671840794621901515, 0,
1.37926999870671840794621901515, 2.86466238235176706687444228967, 3.58706367992406002583482936990, 4.88716868227948365972096587587, 5.32200919003813187335704029210, 6.11525795512597235717276121264, 7.39444563309779465349867586459, 7.56498833244357344695010144284, 8.447381863208563034433050526997
