| L(s) = 1 | + 2·2-s − 2·3-s + 2·4-s − 4·6-s + 3·9-s − 4·12-s + 2·13-s − 4·16-s + 6·18-s + 4·26-s − 4·27-s − 6·31-s − 8·32-s + 6·36-s + 4·37-s − 4·39-s − 16·41-s + 2·43-s + 8·48-s − 5·49-s + 4·52-s − 8·53-s − 8·54-s − 12·62-s − 8·64-s − 6·67-s − 16·71-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 4-s − 1.63·6-s + 9-s − 1.15·12-s + 0.554·13-s − 16-s + 1.41·18-s + 0.784·26-s − 0.769·27-s − 1.07·31-s − 1.41·32-s + 36-s + 0.657·37-s − 0.640·39-s − 2.49·41-s + 0.304·43-s + 1.15·48-s − 5/7·49-s + 0.554·52-s − 1.09·53-s − 1.08·54-s − 1.52·62-s − 64-s − 0.733·67-s − 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.501433473574279825615705143828, −7.81354971406306564487736649230, −7.36165518409022335394046952973, −6.79375443883521821659151920822, −6.40254837576111338701589329363, −6.05569215012514529764032801313, −5.53203277712182766193576425817, −5.15646830785539032141737157736, −4.62309319523854591570832526017, −4.26870432443895851294601186738, −3.45773808162468171855087579517, −3.24466149791278329969031792942, −2.17650184356810029240245516125, −1.42740573370956801547148750389, 0,
1.42740573370956801547148750389, 2.17650184356810029240245516125, 3.24466149791278329969031792942, 3.45773808162468171855087579517, 4.26870432443895851294601186738, 4.62309319523854591570832526017, 5.15646830785539032141737157736, 5.53203277712182766193576425817, 6.05569215012514529764032801313, 6.40254837576111338701589329363, 6.79375443883521821659151920822, 7.36165518409022335394046952973, 7.81354971406306564487736649230, 8.501433473574279825615705143828
