| L(s) = 1 | + 4-s + 6·7-s − 5·9-s + 4·11-s − 2·13-s + 16-s + 6·17-s + 6·25-s + 6·28-s − 10·29-s − 5·36-s − 4·37-s + 8·43-s + 4·44-s + 16·47-s + 13·49-s − 2·52-s − 53-s + 30·59-s − 30·63-s + 64-s + 6·68-s + 24·77-s + 16·81-s − 12·91-s − 4·97-s − 20·99-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 2.26·7-s − 5/3·9-s + 1.20·11-s − 0.554·13-s + 1/4·16-s + 1.45·17-s + 6/5·25-s + 1.13·28-s − 1.85·29-s − 5/6·36-s − 0.657·37-s + 1.21·43-s + 0.603·44-s + 2.33·47-s + 13/7·49-s − 0.277·52-s − 0.137·53-s + 3.90·59-s − 3.77·63-s + 1/8·64-s + 0.727·68-s + 2.73·77-s + 16/9·81-s − 1.25·91-s − 0.406·97-s − 2.01·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056196 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056196 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.148612285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.148612285\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 53 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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
−7.32979161793924210521561267837, −7.25038064772684706053554512263, −6.67987361800666749130348188451, −5.93986309194826935463705061560, −5.75450979320825133654167307588, −5.28069074518443602737935578596, −5.15809734676727653241635712686, −4.53029472544414764179358236482, −3.86057015395193111701613139713, −3.68604192530530606104598929080, −2.93017664736345101722479598160, −2.36672710236137886516193330845, −2.02290526990739871330795391458, −1.28559494094947182426953386776, −0.806120530930692310082326503844,
0.806120530930692310082326503844, 1.28559494094947182426953386776, 2.02290526990739871330795391458, 2.36672710236137886516193330845, 2.93017664736345101722479598160, 3.68604192530530606104598929080, 3.86057015395193111701613139713, 4.53029472544414764179358236482, 5.15809734676727653241635712686, 5.28069074518443602737935578596, 5.75450979320825133654167307588, 5.93986309194826935463705061560, 6.67987361800666749130348188451, 7.25038064772684706053554512263, 7.32979161793924210521561267837
