| L(s) = 1 | − 4·13-s − 8·19-s + 2·25-s − 8·31-s + 4·37-s + 8·43-s + 20·61-s + 8·67-s − 28·73-s − 16·79-s − 28·97-s − 8·103-s + 4·109-s − 10·121-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 + 181-s + ⋯ |
| L(s) = 1 | − 1.10·13-s − 1.83·19-s + 2/5·25-s − 1.43·31-s + 0.657·37-s + 1.21·43-s + 2.56·61-s + 0.977·67-s − 3.27·73-s − 1.80·79-s − 2.84·97-s − 0.788·103-s + 0.383·109-s − 0.909·121-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 + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 166 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.51218116529908702620620166504, −7.48787540797712496743483874060, −7.15266124550141519709892359020, −6.77167896444280890855160561782, −6.24563608894913869400135046650, −6.21512305737288936287229642991, −5.49519651079117864993843431451, −5.37006982049676306500307958175, −5.02114905098027128279915958922, −4.38582871297145479642551825615, −4.08319426141116103943305121375, −4.05548690470664349380117502531, −3.35666084931729769058191730657, −2.79489885101287138119063207250, −2.34445413942131180134481486553, −2.34188766022526201091472767243, −1.50682940868252350405565775611, −1.10739081068999937884634336742, 0, 0,
1.10739081068999937884634336742, 1.50682940868252350405565775611, 2.34188766022526201091472767243, 2.34445413942131180134481486553, 2.79489885101287138119063207250, 3.35666084931729769058191730657, 4.05548690470664349380117502531, 4.08319426141116103943305121375, 4.38582871297145479642551825615, 5.02114905098027128279915958922, 5.37006982049676306500307958175, 5.49519651079117864993843431451, 6.21512305737288936287229642991, 6.24563608894913869400135046650, 6.77167896444280890855160561782, 7.15266124550141519709892359020, 7.48787540797712496743483874060, 7.51218116529908702620620166504
