| L(s) = 1 | + 4·5-s − 38·25-s − 100·29-s + 2·49-s + 188·53-s − 100·73-s − 380·97-s + 380·101-s − 142·121-s − 268·125-s + 127-s + 131-s + 137-s + 139-s − 400·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 338·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 4/5·5-s − 1.51·25-s − 3.44·29-s + 2/49·49-s + 3.54·53-s − 1.36·73-s − 3.91·97-s + 3.76·101-s − 1.17·121-s − 2.14·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 2.75·145-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 2·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.240458850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.240458850\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )( 1 + 10 T + p^{2} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )( 1 + 10 T + p^{2} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 50 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 38 T + p^{2} T^{2} )( 1 + 38 T + p^{2} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 94 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )( 1 + 10 T + p^{2} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 50 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 58 T + p^{2} T^{2} )( 1 + 58 T + p^{2} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 134 T + p^{2} T^{2} )( 1 + 134 T + p^{2} T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 190 T + p^{2} T^{2} )^{2} \) |
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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
−9.252323473804554270994662583211, −8.603258325580597282536607305726, −8.379484714569256146536086506080, −7.65795639840418766304199506872, −7.59664215659282173926497508411, −6.97379956533045215473993456100, −6.93984244779167864714567609113, −5.97133742641685569718494832400, −5.92937407018795759817014023577, −5.54040420567272753428225089478, −5.32174954055330780145144605827, −4.57252794877238667427058945948, −4.09818071117764543335843482563, −3.72107313182437970526660117334, −3.41425095394894885275083059965, −2.52211840332669157915131240029, −2.21548733895160069777541257204, −1.78068829062375822885783506935, −1.19926428104994108417954238602, −0.26688735975999893160163739402,
0.26688735975999893160163739402, 1.19926428104994108417954238602, 1.78068829062375822885783506935, 2.21548733895160069777541257204, 2.52211840332669157915131240029, 3.41425095394894885275083059965, 3.72107313182437970526660117334, 4.09818071117764543335843482563, 4.57252794877238667427058945948, 5.32174954055330780145144605827, 5.54040420567272753428225089478, 5.92937407018795759817014023577, 5.97133742641685569718494832400, 6.93984244779167864714567609113, 6.97379956533045215473993456100, 7.59664215659282173926497508411, 7.65795639840418766304199506872, 8.379484714569256146536086506080, 8.603258325580597282536607305726, 9.252323473804554270994662583211
