| L(s) = 1 | − 2·7-s − 9-s + 12·17-s − 8·23-s + 6·25-s + 16·31-s + 20·41-s + 16·47-s + 3·49-s + 2·63-s + 8·71-s − 4·73-s − 16·79-s + 81-s − 12·89-s + 20·97-s − 28·113-s − 24·119-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 12·153-s + 157-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 1/3·9-s + 2.91·17-s − 1.66·23-s + 6/5·25-s + 2.87·31-s + 3.12·41-s + 2.33·47-s + 3/7·49-s + 0.251·63-s + 0.949·71-s − 0.468·73-s − 1.80·79-s + 1/9·81-s − 1.27·89-s + 2.03·97-s − 2.63·113-s − 2.20·119-s + 2·121-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 − 0.970·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28901376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28901376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.694103890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.694103890\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−8.279397665312073049335373723110, −7.985518312703168641727336835325, −7.65302120166007155431532517894, −7.40671623888464360597444339395, −6.95067797102855960568258859371, −6.52044462824515729705326468352, −5.98982745538633593043346979115, −5.94983842431006207501838394498, −5.60324904344828334472451245663, −5.25411710586797739099366113284, −4.51706386798544310890041215869, −4.35395701484166710581149497524, −3.89614060117994984325577992245, −3.46197395810664827940175595087, −2.88125332952870959049936553264, −2.77799802826630409595912644190, −2.35445385359391739769152291113, −1.47263088493981253453015684547, −0.831924491803121236226692459483, −0.72811319257996027822926713571,
0.72811319257996027822926713571, 0.831924491803121236226692459483, 1.47263088493981253453015684547, 2.35445385359391739769152291113, 2.77799802826630409595912644190, 2.88125332952870959049936553264, 3.46197395810664827940175595087, 3.89614060117994984325577992245, 4.35395701484166710581149497524, 4.51706386798544310890041215869, 5.25411710586797739099366113284, 5.60324904344828334472451245663, 5.94983842431006207501838394498, 5.98982745538633593043346979115, 6.52044462824515729705326468352, 6.95067797102855960568258859371, 7.40671623888464360597444339395, 7.65302120166007155431532517894, 7.985518312703168641727336835325, 8.279397665312073049335373723110
