L(s) = 1 | + 4-s − 7-s − 3·9-s − 3·16-s + 2·25-s − 28-s − 3·36-s − 4·37-s + 8·43-s + 49-s + 3·63-s − 7·64-s − 8·67-s + 16·79-s + 9·81-s + 2·100-s − 4·109-s + 3·112-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 9·144-s − 4·148-s + 149-s + 151-s + ⋯ |
L(s) = 1 | + 1/2·4-s − 0.377·7-s − 9-s − 3/4·16-s + 2/5·25-s − 0.188·28-s − 1/2·36-s − 0.657·37-s + 1.21·43-s + 1/7·49-s + 0.377·63-s − 7/8·64-s − 0.977·67-s + 1.80·79-s + 81-s + 1/5·100-s − 0.383·109-s + 0.283·112-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3/4·144-s − 0.328·148-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7397068074\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7397068074\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_1$ | \( 1 + T \) |
good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 166 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−12.71895350827724469915215914171, −12.10442524035307922634690832050, −11.65864166326525916010083071710, −10.90160972825777864716780762811, −10.65074720861343727188897199409, −9.669715572151983783854790254476, −9.074507625868623172210132263641, −8.519119235509310100736566501089, −7.68338116225197161588634653032, −6.95297562225022579313925127204, −6.27580339173841469061935477107, −5.57679107198489813619663746991, −4.59947820001660719392658595822, −3.39358438112305592317655443101, −2.39777753602607747256751042107,
2.39777753602607747256751042107, 3.39358438112305592317655443101, 4.59947820001660719392658595822, 5.57679107198489813619663746991, 6.27580339173841469061935477107, 6.95297562225022579313925127204, 7.68338116225197161588634653032, 8.519119235509310100736566501089, 9.074507625868623172210132263641, 9.669715572151983783854790254476, 10.65074720861343727188897199409, 10.90160972825777864716780762811, 11.65864166326525916010083071710, 12.10442524035307922634690832050, 12.71895350827724469915215914171