| 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 + 49-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 + ⋯ |
| 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 + 1/7·49-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7749752022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7749752022\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 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
−10.96922517526030591560500511793, −10.26175613408033680908306725927, −10.00137734072722000076745491348, −8.829468246037740149648448974500, −8.679498853252053444462430241617, −8.322081181544343689933636986956, −7.980947296320422939728903060714, −6.91010810779795429087322035266, −6.26051630300961999746999469622, −5.47907564040983295563221596641, −5.29742546009269090365766846421, −4.02723544557718882488682513095, −3.61689003045514298236484607491, −3.04856603410542935889592969685, −1.00806560019316329290903940266,
1.00806560019316329290903940266, 3.04856603410542935889592969685, 3.61689003045514298236484607491, 4.02723544557718882488682513095, 5.29742546009269090365766846421, 5.47907564040983295563221596641, 6.26051630300961999746999469622, 6.91010810779795429087322035266, 7.980947296320422939728903060714, 8.322081181544343689933636986956, 8.679498853252053444462430241617, 8.829468246037740149648448974500, 10.00137734072722000076745491348, 10.26175613408033680908306725927, 10.96922517526030591560500511793
