| L(s) = 1 | + 4·13-s + 12·23-s + 2·25-s − 4·37-s + 20·47-s + 12·49-s − 24·59-s + 20·61-s + 4·71-s − 20·73-s − 8·83-s + 24·97-s + 16·107-s − 20·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 1.10·13-s + 2.50·23-s + 2/5·25-s − 0.657·37-s + 2.91·47-s + 12/7·49-s − 3.12·59-s + 2.56·61-s + 0.474·71-s − 2.34·73-s − 0.878·83-s + 2.43·97-s + 1.54·107-s − 1.91·109-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.07·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.257370792\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.257370792\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 64 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
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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
−8.708939325123522326294394013957, −8.151184771621422146874822353018, −7.69000874155822862744733328026, −7.34189631075212151148084317252, −7.14378782845067483818111445199, −6.73455857713300195719825659145, −6.33330154045577279542136862733, −5.86655015292593048227883369808, −5.68564825947825612859627414756, −5.09075383026184885821874152957, −4.91944239709002797886671611082, −4.37908441592510775091517185739, −3.88778964315032066822358969136, −3.63990661946272883795797310598, −3.12083841645615103625805546097, −2.53062676954705139837227007184, −2.46825761757428208425754601203, −1.30562994308230449742493869779, −1.30307726128232653728848458857, −0.55307771627804534081808136931,
0.55307771627804534081808136931, 1.30307726128232653728848458857, 1.30562994308230449742493869779, 2.46825761757428208425754601203, 2.53062676954705139837227007184, 3.12083841645615103625805546097, 3.63990661946272883795797310598, 3.88778964315032066822358969136, 4.37908441592510775091517185739, 4.91944239709002797886671611082, 5.09075383026184885821874152957, 5.68564825947825612859627414756, 5.86655015292593048227883369808, 6.33330154045577279542136862733, 6.73455857713300195719825659145, 7.14378782845067483818111445199, 7.34189631075212151148084317252, 7.69000874155822862744733328026, 8.151184771621422146874822353018, 8.708939325123522326294394013957
