| L(s) = 1 | − 5-s + 7-s − 4·13-s − 6·17-s + 2·19-s − 6·23-s + 25-s + 2·31-s − 35-s + 2·37-s − 6·41-s − 4·43-s + 49-s + 6·53-s − 12·59-s − 10·61-s + 4·65-s − 4·67-s − 12·71-s − 4·73-s + 8·79-s − 12·83-s + 6·85-s − 6·89-s − 4·91-s − 2·95-s + 8·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 1.10·13-s − 1.45·17-s + 0.458·19-s − 1.25·23-s + 1/5·25-s + 0.359·31-s − 0.169·35-s + 0.328·37-s − 0.937·41-s − 0.609·43-s + 1/7·49-s + 0.824·53-s − 1.56·59-s − 1.28·61-s + 0.496·65-s − 0.488·67-s − 1.42·71-s − 0.468·73-s + 0.900·79-s − 1.31·83-s + 0.650·85-s − 0.635·89-s − 0.419·91-s − 0.205·95-s + 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−9.257333517816083400842117690039, −8.426292162831534806056892454327, −7.65347466823701795715809208185, −6.92647988472169649399702660829, −5.95181539946250990194116023316, −4.82950550974889500271011258263, −4.24036285810516191324072572685, −2.95449156082389576578387745949, −1.85058126503417698669696891278, 0,
1.85058126503417698669696891278, 2.95449156082389576578387745949, 4.24036285810516191324072572685, 4.82950550974889500271011258263, 5.95181539946250990194116023316, 6.92647988472169649399702660829, 7.65347466823701795715809208185, 8.426292162831534806056892454327, 9.257333517816083400842117690039
