| L(s) = 1 | − 9-s − 8·11-s − 8·19-s − 4·29-s + 20·41-s + 14·49-s + 8·59-s + 4·61-s + 16·71-s + 81-s + 12·89-s + 8·99-s − 12·101-s + 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 8·171-s + 173-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 2.41·11-s − 1.83·19-s − 0.742·29-s + 3.12·41-s + 2·49-s + 1.04·59-s + 0.512·61-s + 1.89·71-s + 1/9·81-s + 1.27·89-s + 0.804·99-s − 1.19·101-s + 2.68·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.611·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.569351105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.569351105\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−8.325771227847207151811942807712, −8.140967448783480768884143973670, −7.63496812063886058113374727569, −7.63204457930250470450767922233, −7.00376743580792540946801458869, −6.77499118323635517009114061179, −6.13587021773235790211396173332, −5.86997252422084439175783360476, −5.53929580047940595766000118545, −5.31318203684642177364240039243, −4.65260236706451484675730926985, −4.52946379146492802153149340789, −3.84950428209093844155442976970, −3.72037093931365261792359253374, −2.84606455223297959847265013333, −2.64425551647754646316947954791, −2.17523161323321623188216131784, −2.01769707612024726769949069197, −0.820024293278777442730885082476, −0.43540155510501677890886689459,
0.43540155510501677890886689459, 0.820024293278777442730885082476, 2.01769707612024726769949069197, 2.17523161323321623188216131784, 2.64425551647754646316947954791, 2.84606455223297959847265013333, 3.72037093931365261792359253374, 3.84950428209093844155442976970, 4.52946379146492802153149340789, 4.65260236706451484675730926985, 5.31318203684642177364240039243, 5.53929580047940595766000118545, 5.86997252422084439175783360476, 6.13587021773235790211396173332, 6.77499118323635517009114061179, 7.00376743580792540946801458869, 7.63204457930250470450767922233, 7.63496812063886058113374727569, 8.140967448783480768884143973670, 8.325771227847207151811942807712
