| L(s) = 1 | + 24·3-s + 270·9-s − 424·11-s + 228·17-s − 456·19-s − 194·25-s + 1.08e3·27-s − 1.01e4·33-s − 1.54e3·41-s + 4.76e3·43-s + 4.73e3·49-s + 5.47e3·51-s − 1.09e4·57-s + 2.55e3·59-s + 7.19e3·67-s − 1.95e3·73-s − 4.65e3·75-s − 1.66e4·81-s − 5.32e3·83-s − 1.15e4·89-s + 2.14e4·97-s − 1.14e5·99-s + 3.57e4·107-s − 1.01e4·113-s + 1.05e5·121-s − 3.69e4·123-s + 127-s + ⋯ |
| L(s) = 1 | + 8/3·3-s + 10/3·9-s − 3.50·11-s + 0.788·17-s − 1.26·19-s − 0.310·25-s + 1.48·27-s − 9.34·33-s − 0.916·41-s + 2.57·43-s + 1.97·49-s + 2.10·51-s − 3.36·57-s + 0.733·59-s + 1.60·67-s − 0.367·73-s − 0.827·75-s − 2.53·81-s − 0.772·83-s − 1.45·89-s + 2.28·97-s − 11.6·99-s + 3.12·107-s − 0.791·113-s + 7.20·121-s − 2.44·123-s + 6.20e−5·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(4.100714378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.100714378\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - 4 p T + p^{4} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 194 T^{2} + p^{8} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4738 T^{2} + p^{8} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 212 T + p^{4} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 33406 T^{2} + p^{8} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 114 T + p^{4} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 12 p T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 551938 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1409662 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 1207042 T^{2} + p^{8} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 3149246 T^{2} + p^{8} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 770 T + p^{4} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2380 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 7043458 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 2111938 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 1276 T + p^{4} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 5049602 T^{2} + p^{8} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 3596 T + p^{4} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 242818 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 978 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 178306 T^{2} + p^{8} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 2660 T + p^{4} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 5778 T + p^{4} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10738 T + p^{4} T^{2} )^{2} \) |
| 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
−11.55129796552676276219931639556, −10.77833393379413306308616243280, −10.58141563850199743009086915774, −9.880073685633371916362044758056, −9.847701262565903085612878392142, −8.821455351685500815347020834405, −8.721576550262447274857898328930, −8.203350548316115337578858225804, −7.76940508196235840811616790792, −7.57812005979945319683975292654, −7.07608996893245968630741235871, −5.75467928181782600876554580279, −5.60818383983421296599561268861, −4.71883401224280079937975275807, −4.02038504684383685398982530816, −3.32897592968866774628021454247, −2.80395892238443340349520058781, −2.30604835082519388826984757875, −2.12882384546385250457596493936, −0.50773524821935550324316713775,
0.50773524821935550324316713775, 2.12882384546385250457596493936, 2.30604835082519388826984757875, 2.80395892238443340349520058781, 3.32897592968866774628021454247, 4.02038504684383685398982530816, 4.71883401224280079937975275807, 5.60818383983421296599561268861, 5.75467928181782600876554580279, 7.07608996893245968630741235871, 7.57812005979945319683975292654, 7.76940508196235840811616790792, 8.203350548316115337578858225804, 8.721576550262447274857898328930, 8.821455351685500815347020834405, 9.847701262565903085612878392142, 9.880073685633371916362044758056, 10.58141563850199743009086915774, 10.77833393379413306308616243280, 11.55129796552676276219931639556
