| L(s) = 1 | − 3-s − 2·4-s − 5·7-s + 9-s + 2·12-s − 13-s − 7·19-s + 5·21-s − 7·25-s − 27-s + 10·28-s + 7·31-s − 2·36-s − 5·37-s + 39-s − 8·43-s + 14·49-s + 2·52-s + 7·57-s − 12·61-s − 5·63-s + 8·64-s − 20·67-s − 9·73-s + 7·75-s + 14·76-s − 19·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 1.88·7-s + 1/3·9-s + 0.577·12-s − 0.277·13-s − 1.60·19-s + 1.09·21-s − 7/5·25-s − 0.192·27-s + 1.88·28-s + 1.25·31-s − 1/3·36-s − 0.821·37-s + 0.160·39-s − 1.21·43-s + 2·49-s + 0.277·52-s + 0.927·57-s − 1.53·61-s − 0.629·63-s + 64-s − 2.44·67-s − 1.05·73-s + 0.808·75-s + 1.60·76-s − 2.13·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450387 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450387 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 4 T + p T^{2} ) \) |
| 2383 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 28 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 39 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.308883623231691369778241416269, −7.49884176817991046488962856439, −7.24416753695018923755903919381, −6.41122209365487276045016028051, −6.38509136107731615917691131082, −5.92407111589071152631652165712, −5.29799462062159396087902615450, −4.63669692765275185795834063444, −4.28336841141941620688947032644, −3.80485830614953995125922123896, −3.13741152149491807383546361378, −2.55439166169590667599025805584, −1.57149738905477530259290694318, 0, 0,
1.57149738905477530259290694318, 2.55439166169590667599025805584, 3.13741152149491807383546361378, 3.80485830614953995125922123896, 4.28336841141941620688947032644, 4.63669692765275185795834063444, 5.29799462062159396087902615450, 5.92407111589071152631652165712, 6.38509136107731615917691131082, 6.41122209365487276045016028051, 7.24416753695018923755903919381, 7.49884176817991046488962856439, 8.308883623231691369778241416269
