| L(s) = 1 | + 4-s + 6·5-s + 16-s + 12·17-s + 6·20-s + 17·25-s − 20·37-s − 8·43-s + 24·47-s − 6·59-s + 64-s + 4·67-s + 12·68-s + 10·79-s + 6·80-s + 18·83-s + 72·85-s − 12·89-s + 17·100-s + 12·101-s − 20·109-s − 13·121-s + 18·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 2.68·5-s + 1/4·16-s + 2.91·17-s + 1.34·20-s + 17/5·25-s − 3.28·37-s − 1.21·43-s + 3.50·47-s − 0.781·59-s + 1/8·64-s + 0.488·67-s + 1.45·68-s + 1.12·79-s + 0.670·80-s + 1.97·83-s + 7.80·85-s − 1.27·89-s + 1.69·100-s + 1.19·101-s − 1.91·109-s − 1.18·121-s + 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.637105514\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.637105514\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 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
−8.388380094054281186601061743616, −7.56006473562539600008273040402, −7.49053770083860455326674259398, −6.76340153457321918972152567863, −6.40196296984473786838190696766, −5.88431208298112457486858305913, −5.62582070914781549215591544481, −5.13267260304391847265082866628, −5.10036072121782338740270472262, −3.69829470531395738272962092099, −3.52994453436524682105467711302, −2.65191870625457881403725845166, −2.28166780967037125049719495570, −1.52906039199785314571915081352, −1.22415852747718716380535497945,
1.22415852747718716380535497945, 1.52906039199785314571915081352, 2.28166780967037125049719495570, 2.65191870625457881403725845166, 3.52994453436524682105467711302, 3.69829470531395738272962092099, 5.10036072121782338740270472262, 5.13267260304391847265082866628, 5.62582070914781549215591544481, 5.88431208298112457486858305913, 6.40196296984473786838190696766, 6.76340153457321918972152567863, 7.49053770083860455326674259398, 7.56006473562539600008273040402, 8.388380094054281186601061743616
