L(s) = 1 | + 3·5-s + 2·9-s + 3·13-s − 8·17-s + 2·25-s − 12·29-s − 12·37-s + 8·41-s + 6·45-s + 6·49-s + 20·53-s − 4·61-s + 9·65-s + 4·73-s − 5·81-s − 24·85-s − 4·89-s − 16·97-s + 12·101-s + 8·109-s + 6·117-s − 14·121-s − 10·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
L(s) = 1 | + 1.34·5-s + 2/3·9-s + 0.832·13-s − 1.94·17-s + 2/5·25-s − 2.22·29-s − 1.97·37-s + 1.24·41-s + 0.894·45-s + 6/7·49-s + 2.74·53-s − 0.512·61-s + 1.11·65-s + 0.468·73-s − 5/9·81-s − 2.60·85-s − 0.423·89-s − 1.62·97-s + 1.19·101-s + 0.766·109-s + 0.554·117-s − 1.27·121-s − 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16640 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16640 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.373707389\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.373707389\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.99665603084136933695777268158, −10.43829028140359191111677736213, −10.03940461951279911716435138892, −9.253044758571529078077564771140, −9.025900933301165468584956072423, −8.525701822214155923531559284537, −7.52200466865104385187389079926, −7.00670577049561146871013475993, −6.46357626140388597733074399726, −5.72262240981459708464889979455, −5.37694307459617484012203042109, −4.29399920891880282775072036850, −3.76502980897935895002989580605, −2.38569504525986996554649613386, −1.73316942140065786408528891368,
1.73316942140065786408528891368, 2.38569504525986996554649613386, 3.76502980897935895002989580605, 4.29399920891880282775072036850, 5.37694307459617484012203042109, 5.72262240981459708464889979455, 6.46357626140388597733074399726, 7.00670577049561146871013475993, 7.52200466865104385187389079926, 8.525701822214155923531559284537, 9.025900933301165468584956072423, 9.253044758571529078077564771140, 10.03940461951279911716435138892, 10.43829028140359191111677736213, 10.99665603084136933695777268158