| L(s) = 1 | − 2·5-s − 9-s − 8·11-s − 25-s + 12·29-s − 8·31-s + 20·41-s + 2·45-s + 16·55-s + 8·59-s − 4·61-s + 24·79-s + 81-s − 20·89-s + 8·99-s + 4·101-s + 4·109-s + 26·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s − 24·145-s + 149-s + 151-s + 16·155-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1/3·9-s − 2.41·11-s − 1/5·25-s + 2.22·29-s − 1.43·31-s + 3.12·41-s + 0.298·45-s + 2.15·55-s + 1.04·59-s − 0.512·61-s + 2.70·79-s + 1/9·81-s − 2.11·89-s + 0.804·99-s + 0.398·101-s + 0.383·109-s + 2.36·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.99·145-s + 0.0819·149-s + 0.0813·151-s + 1.28·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8643600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8643600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.035208664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.035208664\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.738659136578894848488510340048, −8.515115661106472199158030756414, −8.111349951585440869201714362962, −7.77839136969694624616842462399, −7.49930856919394887975895744223, −7.37719524432752333665482498017, −6.57169978731651954784374164866, −6.41051605657725976624063571491, −5.64312235629762590557318185782, −5.59213847209174319582998412860, −5.05627106343126384645776604662, −4.73641686312773950260903337717, −4.12099934700969364825161472499, −3.96588714301193085480673723765, −3.13662821620282539175315932176, −2.91083144254592597922393623083, −2.43433062179563664002280813199, −2.02860675558636914170257481991, −0.949879953779989810366524534503, −0.38912957418605204233968045201,
0.38912957418605204233968045201, 0.949879953779989810366524534503, 2.02860675558636914170257481991, 2.43433062179563664002280813199, 2.91083144254592597922393623083, 3.13662821620282539175315932176, 3.96588714301193085480673723765, 4.12099934700969364825161472499, 4.73641686312773950260903337717, 5.05627106343126384645776604662, 5.59213847209174319582998412860, 5.64312235629762590557318185782, 6.41051605657725976624063571491, 6.57169978731651954784374164866, 7.37719524432752333665482498017, 7.49930856919394887975895744223, 7.77839136969694624616842462399, 8.111349951585440869201714362962, 8.515115661106472199158030756414, 8.738659136578894848488510340048
