| L(s) = 1 | + 2·4-s − 8·13-s − 2·19-s − 4·25-s − 14·31-s + 16·37-s − 2·43-s − 16·52-s + 10·61-s − 8·64-s + 4·67-s − 2·73-s − 4·76-s − 8·79-s − 2·97-s − 8·100-s + 4·103-s − 2·109-s + 2·121-s − 28·124-s + 127-s + 131-s + 137-s + 139-s + 32·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 4-s − 2.21·13-s − 0.458·19-s − 4/5·25-s − 2.51·31-s + 2.63·37-s − 0.304·43-s − 2.21·52-s + 1.28·61-s − 64-s + 0.488·67-s − 0.234·73-s − 0.458·76-s − 0.900·79-s − 0.203·97-s − 4/5·100-s + 0.394·103-s − 0.191·109-s + 2/11·121-s − 2.51·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.63·148-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.578897573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.578897573\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 100 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 172 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 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
−9.849125313951014007785093658268, −9.452144102853413527653525586633, −9.228577521888416344611621064031, −8.581928299849798844515665283663, −8.105138752059572646524517237748, −7.56076006955269034205658484004, −7.50585473311539148064127254111, −6.92468644493538947964879493021, −6.77713073850347960008532016184, −6.09688137784502721804176671179, −5.60689318223731086794296073385, −5.43103351972531468928767838787, −4.51509961686787975786451916627, −4.50834285369947658073353825069, −3.75018725417803188042230433155, −3.10103467475367051351046320982, −2.55380775165440784502363522246, −2.14379640500151833657949044127, −1.75514751078195191206685243181, −0.47867821221014889538509864765,
0.47867821221014889538509864765, 1.75514751078195191206685243181, 2.14379640500151833657949044127, 2.55380775165440784502363522246, 3.10103467475367051351046320982, 3.75018725417803188042230433155, 4.50834285369947658073353825069, 4.51509961686787975786451916627, 5.43103351972531468928767838787, 5.60689318223731086794296073385, 6.09688137784502721804176671179, 6.77713073850347960008532016184, 6.92468644493538947964879493021, 7.50585473311539148064127254111, 7.56076006955269034205658484004, 8.105138752059572646524517237748, 8.581928299849798844515665283663, 9.228577521888416344611621064031, 9.452144102853413527653525586633, 9.849125313951014007785093658268
