| L(s) = 1 | + 2·3-s + 3·9-s − 8·19-s − 10·25-s + 4·27-s + 12·29-s + 16·31-s − 4·37-s + 16·47-s + 4·53-s − 16·57-s − 8·59-s − 20·75-s + 5·81-s + 24·83-s + 24·87-s + 32·93-s + 16·103-s + 12·109-s − 8·111-s + 4·113-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 32·141-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s − 1.83·19-s − 2·25-s + 0.769·27-s + 2.22·29-s + 2.87·31-s − 0.657·37-s + 2.33·47-s + 0.549·53-s − 2.11·57-s − 1.04·59-s − 2.30·75-s + 5/9·81-s + 2.63·83-s + 2.57·87-s + 3.31·93-s + 1.57·103-s + 1.14·109-s − 0.759·111-s + 0.376·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.69·141-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.833796394\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.833796394\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 146 T^{2} + p^{2} T^{4} \) |
| 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
−7.911933362660355096301638688843, −7.67753911174006134563695587785, −7.30051105559858652801059203182, −6.82672441438608402066648718749, −6.40322208328912833402246525687, −6.33209716339531137071855047537, −6.02751448165583504117786541015, −5.48216527177464350845805590794, −4.94112180930035082106382725068, −4.63340131782117377259318855332, −4.32562635880199084549945882466, −4.01754468724643796816698474815, −3.72093086885127854385664539478, −3.10873618733757644875261117089, −2.75346007989027902441743287998, −2.51021804242624827538827399585, −1.95245090885116493869228210470, −1.78724827875133163800567208756, −0.857457788768882232748049337839, −0.60888953459971999994697655639,
0.60888953459971999994697655639, 0.857457788768882232748049337839, 1.78724827875133163800567208756, 1.95245090885116493869228210470, 2.51021804242624827538827399585, 2.75346007989027902441743287998, 3.10873618733757644875261117089, 3.72093086885127854385664539478, 4.01754468724643796816698474815, 4.32562635880199084549945882466, 4.63340131782117377259318855332, 4.94112180930035082106382725068, 5.48216527177464350845805590794, 6.02751448165583504117786541015, 6.33209716339531137071855047537, 6.40322208328912833402246525687, 6.82672441438608402066648718749, 7.30051105559858652801059203182, 7.67753911174006134563695587785, 7.911933362660355096301638688843
