L(s) = 1 | + 4-s + 9-s − 4·11-s + 2·13-s − 3·16-s − 2·17-s − 4·19-s − 2·25-s + 8·31-s + 36-s − 14·37-s − 4·44-s − 8·47-s − 49-s + 2·52-s − 7·64-s − 2·68-s − 12·71-s − 4·76-s + 81-s − 18·89-s − 4·99-s − 2·100-s + 10·101-s + 2·109-s − 2·113-s + 2·117-s + ⋯ |
L(s) = 1 | + 1/2·4-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 3/4·16-s − 0.485·17-s − 0.917·19-s − 2/5·25-s + 1.43·31-s + 1/6·36-s − 2.30·37-s − 0.603·44-s − 1.16·47-s − 1/7·49-s + 0.277·52-s − 7/8·64-s − 0.242·68-s − 1.42·71-s − 0.458·76-s + 1/9·81-s − 1.90·89-s − 0.402·99-s − 1/5·100-s + 0.995·101-s + 0.191·109-s − 0.188·113-s + 0.184·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221841 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221841 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 157 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 65 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 61 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 138 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 134 T^{2} + p^{2} T^{4} \) |
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
−8.600880989554269961839054807695, −8.466334499617947271307519615289, −7.82518434363817597256867507777, −7.35475260706879974578016860262, −6.74482293091353902445130051099, −6.54988525104159795299261648224, −5.90507484735023464570841611575, −5.35474862672613164845552761341, −4.68761726172784495564637266647, −4.37084941620788759664953324954, −3.53457398354938014465723819771, −2.89319909526715845526126540440, −2.23534806636514928119903856201, −1.59131130085401414686637681654, 0,
1.59131130085401414686637681654, 2.23534806636514928119903856201, 2.89319909526715845526126540440, 3.53457398354938014465723819771, 4.37084941620788759664953324954, 4.68761726172784495564637266647, 5.35474862672613164845552761341, 5.90507484735023464570841611575, 6.54988525104159795299261648224, 6.74482293091353902445130051099, 7.35475260706879974578016860262, 7.82518434363817597256867507777, 8.466334499617947271307519615289, 8.600880989554269961839054807695