| L(s) = 1 | − 2·4-s − 7-s + 4·16-s + 5·25-s + 2·28-s − 2·37-s − 8·43-s − 6·49-s − 8·64-s − 5·67-s + 34·79-s − 10·100-s − 17·109-s − 4·112-s − 11·121-s + 127-s + 131-s + 137-s + 139-s + 4·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + 16·172-s + ⋯ |
| L(s) = 1 | − 4-s − 0.377·7-s + 16-s + 25-s + 0.377·28-s − 0.328·37-s − 1.21·43-s − 6/7·49-s − 64-s − 0.610·67-s + 3.82·79-s − 100-s − 1.62·109-s − 0.377·112-s − 121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.328·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s + 1.21·172-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−7.77883679752783437419350037539, −7.05430437262565829676467793042, −6.75234966276754939307987070763, −6.24649143062360731698773583503, −5.90568531386463437730901920207, −5.15202182182298009042952582324, −5.02653691721674116911538805414, −4.60555074604065345490349155922, −3.89478168182631749046179766128, −3.56515433226044781608492768941, −3.09476556864331292870122064439, −2.46105089709792716214081542587, −1.65554483967698566697270220481, −0.921790063783651499399836152843, 0,
0.921790063783651499399836152843, 1.65554483967698566697270220481, 2.46105089709792716214081542587, 3.09476556864331292870122064439, 3.56515433226044781608492768941, 3.89478168182631749046179766128, 4.60555074604065345490349155922, 5.02653691721674116911538805414, 5.15202182182298009042952582324, 5.90568531386463437730901920207, 6.24649143062360731698773583503, 6.75234966276754939307987070763, 7.05430437262565829676467793042, 7.77883679752783437419350037539
