| L(s) = 1 | − 2·2-s − 4-s + 7-s + 8·8-s − 2·14-s − 7·16-s − 16·23-s + 25-s − 28-s + 4·29-s − 14·32-s − 4·37-s + 8·43-s + 32·46-s + 49-s − 2·50-s − 20·53-s + 8·56-s − 8·58-s + 35·64-s + 8·67-s + 24·71-s + 8·74-s + 16·79-s − 16·86-s + 16·92-s − 2·98-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1/2·4-s + 0.377·7-s + 2.82·8-s − 0.534·14-s − 7/4·16-s − 3.33·23-s + 1/5·25-s − 0.188·28-s + 0.742·29-s − 2.47·32-s − 0.657·37-s + 1.21·43-s + 4.71·46-s + 1/7·49-s − 0.282·50-s − 2.74·53-s + 1.06·56-s − 1.05·58-s + 35/8·64-s + 0.977·67-s + 2.84·71-s + 0.929·74-s + 1.80·79-s − 1.72·86-s + 1.66·92-s − 0.202·98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 694575 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 694575 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.261177977090796325600765542566, −7.82733728798443659692174656726, −7.73831918709431562684545235915, −6.82600380115164878858049675491, −6.48451110526696282246961711325, −5.81605182603986018580602603999, −5.25448844749250196408410433194, −4.87484917945516723546545158294, −4.17084275561163201389021211529, −4.05504421439464721652879145719, −3.34833988846054044432199186298, −2.17599002568151719086793905114, −1.79849192746842450415563444508, −0.888154170390839902230274080915, 0,
0.888154170390839902230274080915, 1.79849192746842450415563444508, 2.17599002568151719086793905114, 3.34833988846054044432199186298, 4.05504421439464721652879145719, 4.17084275561163201389021211529, 4.87484917945516723546545158294, 5.25448844749250196408410433194, 5.81605182603986018580602603999, 6.48451110526696282246961711325, 6.82600380115164878858049675491, 7.73831918709431562684545235915, 7.82733728798443659692174656726, 8.261177977090796325600765542566
