| L(s) = 1 | − 2·2-s + 3·4-s − 4·7-s − 4·8-s + 8·14-s + 5·16-s − 25-s − 12·28-s + 18·29-s − 6·32-s − 2·37-s + 16·43-s + 9·49-s + 2·50-s − 12·53-s + 16·56-s − 36·58-s + 7·64-s − 8·67-s − 24·71-s + 4·74-s − 32·79-s − 32·86-s − 18·98-s − 3·100-s + 24·106-s + 24·107-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.51·7-s − 1.41·8-s + 2.13·14-s + 5/4·16-s − 1/5·25-s − 2.26·28-s + 3.34·29-s − 1.06·32-s − 0.328·37-s + 2.43·43-s + 9/7·49-s + 0.282·50-s − 1.64·53-s + 2.13·56-s − 4.72·58-s + 7/8·64-s − 0.977·67-s − 2.84·71-s + 0.464·74-s − 3.60·79-s − 3.45·86-s − 1.81·98-s − 0.299·100-s + 2.33·106-s + 2.32·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.83804568267963926704994118011, −7.23037692302101024142168718583, −7.16277862972928710155421440227, −6.39717818377493347523565938673, −6.29261335708839240986736404399, −5.90045021115349717827393199661, −5.27862826325532129316709233362, −4.37202457039453886575368172384, −4.25638665319375785757127060430, −3.16355858187330818291475189494, −2.94651413139897301734292877776, −2.59312842582062058372859748021, −1.58788487890578855994666762814, −0.901297860986010122087650764417, 0,
0.901297860986010122087650764417, 1.58788487890578855994666762814, 2.59312842582062058372859748021, 2.94651413139897301734292877776, 3.16355858187330818291475189494, 4.25638665319375785757127060430, 4.37202457039453886575368172384, 5.27862826325532129316709233362, 5.90045021115349717827393199661, 6.29261335708839240986736404399, 6.39717818377493347523565938673, 7.16277862972928710155421440227, 7.23037692302101024142168718583, 7.83804568267963926704994118011
