L(s) = 1 | + 2·5-s − 8·11-s − 6·13-s − 12·17-s + 8·19-s + 2·25-s + 6·29-s + 8·31-s − 2·37-s + 8·43-s − 16·47-s − 2·49-s − 14·53-s − 16·55-s + 6·61-s − 12·65-s − 16·67-s − 24·79-s + 8·83-s − 24·85-s + 16·95-s + 16·97-s − 14·101-s + 16·107-s − 10·109-s − 32·113-s + 32·121-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 2.41·11-s − 1.66·13-s − 2.91·17-s + 1.83·19-s + 2/5·25-s + 1.11·29-s + 1.43·31-s − 0.328·37-s + 1.21·43-s − 2.33·47-s − 2/7·49-s − 1.92·53-s − 2.15·55-s + 0.768·61-s − 1.48·65-s − 1.95·67-s − 2.70·79-s + 0.878·83-s − 2.60·85-s + 1.64·95-s + 1.62·97-s − 1.39·101-s + 1.54·107-s − 0.957·109-s − 3.01·113-s + 2.90·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.09689716198\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09689716198\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−8.362017209381235198744161741666, −8.192103139030257720776910959923, −7.76893024660610000797348726281, −7.36856164053651633610227440518, −7.15719406853020923425070024553, −6.55170528675756967593007403050, −6.48848227074242941726358182996, −5.92149449244444590145191603342, −5.42199816231411694767548020303, −5.20742374514139426483553794644, −4.73293639549165470801873039382, −4.58338144069477121829593433729, −4.33951796653725549354922179811, −3.10016365707080411445844424290, −3.08006260132763794948884773551, −2.65991354410381276931292855469, −2.24008687379058077700155885649, −1.91308135062215058560307678058, −1.11624214489511659905299625634, −0.085234374259395870710813123921,
0.085234374259395870710813123921, 1.11624214489511659905299625634, 1.91308135062215058560307678058, 2.24008687379058077700155885649, 2.65991354410381276931292855469, 3.08006260132763794948884773551, 3.10016365707080411445844424290, 4.33951796653725549354922179811, 4.58338144069477121829593433729, 4.73293639549165470801873039382, 5.20742374514139426483553794644, 5.42199816231411694767548020303, 5.92149449244444590145191603342, 6.48848227074242941726358182996, 6.55170528675756967593007403050, 7.15719406853020923425070024553, 7.36856164053651633610227440518, 7.76893024660610000797348726281, 8.192103139030257720776910959923, 8.362017209381235198744161741666