L(s) = 1 | − 2-s − 4-s + 3·8-s − 2·9-s − 7·11-s − 16-s − 8·17-s + 2·18-s + 6·19-s + 7·22-s − 6·25-s − 3·27-s − 5·32-s + 8·34-s + 2·36-s − 6·38-s − 2·41-s + 4·43-s + 7·44-s − 4·49-s + 6·50-s + 3·54-s + 7·64-s + 5·67-s + 8·68-s − 6·72-s − 9·73-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s − 2/3·9-s − 2.11·11-s − 1/4·16-s − 1.94·17-s + 0.471·18-s + 1.37·19-s + 1.49·22-s − 6/5·25-s − 0.577·27-s − 0.883·32-s + 1.37·34-s + 1/3·36-s − 0.973·38-s − 0.312·41-s + 0.609·43-s + 1.05·44-s − 4/7·49-s + 0.848·50-s + 0.408·54-s + 7/8·64-s + 0.610·67-s + 0.970·68-s − 0.707·72-s − 1.05·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15936 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15936 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 10 T + p T^{2} ) \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 51 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 85 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 87 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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
−10.74433800535156877153979848015, −10.16518672559382861769867758916, −9.716319197810368213029190518222, −9.186122526627509416090619069088, −8.517570365913117989274421248918, −8.169217583110402994245497503336, −7.53159554253679515255920073744, −7.15337561951127586942226377650, −6.04400141283971692535889787756, −5.43681516396191237807406769919, −4.89735903256202936455630169262, −4.12328357885593378890854010587, −3.01948987046768835443838392986, −2.10644602273849325777138561407, 0,
2.10644602273849325777138561407, 3.01948987046768835443838392986, 4.12328357885593378890854010587, 4.89735903256202936455630169262, 5.43681516396191237807406769919, 6.04400141283971692535889787756, 7.15337561951127586942226377650, 7.53159554253679515255920073744, 8.169217583110402994245497503336, 8.517570365913117989274421248918, 9.186122526627509416090619069088, 9.716319197810368213029190518222, 10.16518672559382861769867758916, 10.74433800535156877153979848015