| L(s) = 1 | + 2·5-s − 9-s − 6·11-s − 25-s + 8·29-s − 16·31-s − 6·41-s − 2·45-s + 13·49-s − 12·55-s − 8·59-s + 2·61-s + 18·71-s − 26·79-s + 81-s − 6·89-s + 6·99-s + 28·101-s + 5·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 16·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1/3·9-s − 1.80·11-s − 1/5·25-s + 1.48·29-s − 2.87·31-s − 0.937·41-s − 0.298·45-s + 13/7·49-s − 1.61·55-s − 1.04·59-s + 0.256·61-s + 2.13·71-s − 2.92·79-s + 1/9·81-s − 0.635·89-s + 0.603·99-s + 2.78·101-s + 5/11·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.32·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.199453364\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.199453364\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 49 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 167 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.773728636133080312689235350611, −8.573755865672836014941425175368, −8.219853200684959666778929705441, −7.59224557660591393123620365164, −7.47582324786870707005568503919, −7.15523752216446281476237824883, −6.44692268041903853478090096510, −6.31427423566177343679936023938, −5.66142056884544516090803070266, −5.40713394923726595656690049580, −5.28394636175809552372994800386, −4.77246378260977190178259069080, −4.18601052721572370724692571204, −3.75890643091260885501694609157, −3.11841546705489011220588882939, −2.85215754336118841052723502971, −2.18841125457261721178822716678, −2.02637521435282986737915738601, −1.26310360948669339854748590139, −0.33429194401949976759119696875,
0.33429194401949976759119696875, 1.26310360948669339854748590139, 2.02637521435282986737915738601, 2.18841125457261721178822716678, 2.85215754336118841052723502971, 3.11841546705489011220588882939, 3.75890643091260885501694609157, 4.18601052721572370724692571204, 4.77246378260977190178259069080, 5.28394636175809552372994800386, 5.40713394923726595656690049580, 5.66142056884544516090803070266, 6.31427423566177343679936023938, 6.44692268041903853478090096510, 7.15523752216446281476237824883, 7.47582324786870707005568503919, 7.59224557660591393123620365164, 8.219853200684959666778929705441, 8.573755865672836014941425175368, 8.773728636133080312689235350611
