| L(s) = 1 | + 4·9-s + 8·13-s − 6·17-s + 8·19-s + 2·25-s + 16·43-s + 24·47-s − 12·53-s − 8·67-s + 7·81-s − 24·89-s − 24·101-s − 16·103-s + 32·117-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + 157-s + 163-s + 167-s + 22·169-s + 32·171-s + ⋯ |
| L(s) = 1 | + 4/3·9-s + 2.21·13-s − 1.45·17-s + 1.83·19-s + 2/5·25-s + 2.43·43-s + 3.50·47-s − 1.64·53-s − 0.977·67-s + 7/9·81-s − 2.54·89-s − 2.38·101-s − 1.57·103-s + 2.95·117-s + 1.81·121-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 − 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 2.44·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.073961275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.073961275\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 92 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 140 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−9.171924702527574337555717665426, −8.478853418615852924070730515768, −8.117150631464355883057345796486, −7.58244010693781146472357456071, −7.30460009731033303507926265645, −6.98330175918415174496554911145, −6.71585799606762800398881691780, −5.99073947578360479849892432441, −5.92416336827612259671591408493, −5.55971352873620900131879208340, −4.94533604186819072799102090721, −4.33556329033103752053931528617, −4.23651806592263843561796399330, −3.84723019786387171980301979499, −3.35557915310758406773997657842, −2.73350639811331207734201520511, −2.40401879973578705823224118485, −1.39188626362306616951778318912, −1.37450596698300350519268658808, −0.69197666760147965336248626507,
0.69197666760147965336248626507, 1.37450596698300350519268658808, 1.39188626362306616951778318912, 2.40401879973578705823224118485, 2.73350639811331207734201520511, 3.35557915310758406773997657842, 3.84723019786387171980301979499, 4.23651806592263843561796399330, 4.33556329033103752053931528617, 4.94533604186819072799102090721, 5.55971352873620900131879208340, 5.92416336827612259671591408493, 5.99073947578360479849892432441, 6.71585799606762800398881691780, 6.98330175918415174496554911145, 7.30460009731033303507926265645, 7.58244010693781146472357456071, 8.117150631464355883057345796486, 8.478853418615852924070730515768, 9.171924702527574337555717665426
