| L(s) = 1 | − 5-s − 5·11-s + 3·17-s + 5·19-s + 6·23-s + 25-s − 10·29-s − 2·31-s + 4·37-s − 3·41-s + 3·43-s + 4·47-s − 7·49-s − 6·53-s + 5·55-s − 3·59-s + 2·61-s − 11·67-s − 14·71-s − 15·73-s + 10·79-s − 12·83-s − 3·85-s + 14·89-s − 5·95-s − 13·97-s − 12·101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.50·11-s + 0.727·17-s + 1.14·19-s + 1.25·23-s + 1/5·25-s − 1.85·29-s − 0.359·31-s + 0.657·37-s − 0.468·41-s + 0.457·43-s + 0.583·47-s − 49-s − 0.824·53-s + 0.674·55-s − 0.390·59-s + 0.256·61-s − 1.34·67-s − 1.66·71-s − 1.75·73-s + 1.12·79-s − 1.31·83-s − 0.325·85-s + 1.48·89-s − 0.512·95-s − 1.31·97-s − 1.19·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−8.064414615847267775374623395329, −7.58866790297414207793875098592, −7.05680149447394018089593204064, −5.75961548237591321792684367464, −5.32887920577483785715532145847, −4.45856464210270908729341587586, −3.33978695832999828223955734190, −2.77978308205801535419241515682, −1.40994869662911522204300069715, 0,
1.40994869662911522204300069715, 2.77978308205801535419241515682, 3.33978695832999828223955734190, 4.45856464210270908729341587586, 5.32887920577483785715532145847, 5.75961548237591321792684367464, 7.05680149447394018089593204064, 7.58866790297414207793875098592, 8.064414615847267775374623395329
