| L(s) = 1 | − 2·2-s + 3·4-s − 7-s − 4·8-s − 5·9-s + 6·11-s + 2·14-s + 5·16-s + 10·18-s − 12·22-s − 3·28-s − 12·29-s − 6·32-s − 15·36-s − 16·37-s − 16·43-s + 18·44-s + 49-s + 24·53-s + 4·56-s + 24·58-s + 5·63-s + 7·64-s + 14·67-s + 12·71-s + 20·72-s + 32·74-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.377·7-s − 1.41·8-s − 5/3·9-s + 1.80·11-s + 0.534·14-s + 5/4·16-s + 2.35·18-s − 2.55·22-s − 0.566·28-s − 2.22·29-s − 1.06·32-s − 5/2·36-s − 2.63·37-s − 2.43·43-s + 2.71·44-s + 1/7·49-s + 3.29·53-s + 0.534·56-s + 3.15·58-s + 0.629·63-s + 7/8·64-s + 1.71·67-s + 1.42·71-s + 2.35·72-s + 3.71·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 857500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 857500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.241263185170336936107387248232, −7.62583420866028467275275698765, −7.02864082573576697328686349834, −6.80760429487312719242044771491, −6.40152272675796392235796567725, −5.91589072886351065562126787974, −5.26770803610804114861303773339, −5.17711518617506807839806183328, −3.76060769659553506276694871497, −3.69265834103033057215511968857, −3.22985803075868355514464176224, −2.12269688406492296074462119899, −2.02089552612607080439032267478, −0.926909876778585217976162918755, 0,
0.926909876778585217976162918755, 2.02089552612607080439032267478, 2.12269688406492296074462119899, 3.22985803075868355514464176224, 3.69265834103033057215511968857, 3.76060769659553506276694871497, 5.17711518617506807839806183328, 5.26770803610804114861303773339, 5.91589072886351065562126787974, 6.40152272675796392235796567725, 6.80760429487312719242044771491, 7.02864082573576697328686349834, 7.62583420866028467275275698765, 8.241263185170336936107387248232
