| L(s) = 1 | + 3·4-s + 5·16-s − 12·23-s − 6·25-s − 2·31-s + 12·37-s + 6·47-s − 8·49-s − 18·53-s + 3·64-s − 10·67-s + 6·71-s − 6·89-s − 36·92-s + 4·97-s − 18·100-s − 8·103-s − 24·113-s − 6·124-s + 127-s + 131-s + 137-s + 139-s + 36·148-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 5/4·16-s − 2.50·23-s − 6/5·25-s − 0.359·31-s + 1.97·37-s + 0.875·47-s − 8/7·49-s − 2.47·53-s + 3/8·64-s − 1.22·67-s + 0.712·71-s − 0.635·89-s − 3.75·92-s + 0.406·97-s − 9/5·100-s − 0.788·103-s − 2.25·113-s − 0.538·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.95·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.87307010068916894712436178554, −7.48014103642367909133385954010, −6.91306174116807133430022371591, −6.34187756656271942114130166001, −6.17252423983495797378995859342, −5.84373326086234332449384753680, −5.29769388162667777748987536255, −4.51951720623969887114983375630, −4.12470925677134161887765289386, −3.59850678568799535273463117016, −2.94680035427505770998607255289, −2.42913427837582974044160843893, −1.91144874927117848781753431354, −1.40820629231913257434192264125, 0,
1.40820629231913257434192264125, 1.91144874927117848781753431354, 2.42913427837582974044160843893, 2.94680035427505770998607255289, 3.59850678568799535273463117016, 4.12470925677134161887765289386, 4.51951720623969887114983375630, 5.29769388162667777748987536255, 5.84373326086234332449384753680, 6.17252423983495797378995859342, 6.34187756656271942114130166001, 6.91306174116807133430022371591, 7.48014103642367909133385954010, 7.87307010068916894712436178554
