| L(s) = 1 | + 2·9-s + 12·17-s − 12·41-s − 14·49-s − 4·73-s − 5·81-s + 36·89-s − 20·97-s − 36·113-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 2/3·9-s + 2.91·17-s − 1.87·41-s − 2·49-s − 0.468·73-s − 5/9·81-s + 3.81·89-s − 2.03·97-s − 3.38·113-s − 1.27·121-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 + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.537997386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.537997386\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 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
−9.694416774443088210999715477646, −9.313892019799464381556970285465, −8.851616209863958700260998977023, −8.172631021432919553864616120885, −8.103774700995373890607445658339, −7.60788838758082033607254322932, −7.37550334194041508567538602736, −6.68281195308945160314481488920, −6.50593193379168764596163133492, −5.95307842161694148323001762585, −5.28142301435937642599599931765, −5.26516734667149971724317250711, −4.74599182667682973572692993289, −3.92857836024839309430758945156, −3.77096149761736834639654490818, −2.98866927850474552144855626409, −2.94778991351403219971777831753, −1.65869642098408123024634693077, −1.58120374473771042556784119673, −0.65002429655742763620723412684,
0.65002429655742763620723412684, 1.58120374473771042556784119673, 1.65869642098408123024634693077, 2.94778991351403219971777831753, 2.98866927850474552144855626409, 3.77096149761736834639654490818, 3.92857836024839309430758945156, 4.74599182667682973572692993289, 5.26516734667149971724317250711, 5.28142301435937642599599931765, 5.95307842161694148323001762585, 6.50593193379168764596163133492, 6.68281195308945160314481488920, 7.37550334194041508567538602736, 7.60788838758082033607254322932, 8.103774700995373890607445658339, 8.172631021432919553864616120885, 8.851616209863958700260998977023, 9.313892019799464381556970285465, 9.694416774443088210999715477646
