| L(s) = 1 | − 2·4-s + 2·5-s − 5·9-s − 10·13-s + 4·16-s − 6·17-s − 4·20-s + 3·25-s + 6·29-s + 10·36-s + 4·37-s + 24·41-s − 10·45-s + 20·52-s + 24·53-s − 16·61-s − 8·64-s − 20·65-s + 12·68-s − 4·73-s + 8·80-s + 16·81-s − 12·85-s + 24·89-s + 2·97-s − 6·100-s − 12·101-s + ⋯ |
| L(s) = 1 | − 4-s + 0.894·5-s − 5/3·9-s − 2.77·13-s + 16-s − 1.45·17-s − 0.894·20-s + 3/5·25-s + 1.11·29-s + 5/3·36-s + 0.657·37-s + 3.74·41-s − 1.49·45-s + 2.77·52-s + 3.29·53-s − 2.04·61-s − 64-s − 2.48·65-s + 1.45·68-s − 0.468·73-s + 0.894·80-s + 16/9·81-s − 1.30·85-s + 2.54·89-s + 0.203·97-s − 3/5·100-s − 1.19·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
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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
−7.80156257853413862136981277007, −7.59679103544400835112633947355, −7.18840908028866321022100981317, −6.30078072165756181986797640838, −6.23576189288857341841080164821, −5.51580216499301883688612135353, −5.30121167828360660438204787375, −4.66901616031576415432379359350, −4.48390865811726709341858080699, −3.81894678634232741164227871759, −2.72282955438852544862637314814, −2.61198254943438177491615875407, −2.28641578958231941287903879491, −0.863841609060871512635709852891, 0,
0.863841609060871512635709852891, 2.28641578958231941287903879491, 2.61198254943438177491615875407, 2.72282955438852544862637314814, 3.81894678634232741164227871759, 4.48390865811726709341858080699, 4.66901616031576415432379359350, 5.30121167828360660438204787375, 5.51580216499301883688612135353, 6.23576189288857341841080164821, 6.30078072165756181986797640838, 7.18840908028866321022100981317, 7.59679103544400835112633947355, 7.80156257853413862136981277007
