| L(s) = 1 | − 2·4-s + 2·9-s + 12·11-s + 4·16-s − 5·25-s − 4·36-s − 24·44-s − 7·49-s − 8·64-s − 5·81-s + 24·99-s + 10·100-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 8·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + 48·176-s + 179-s + ⋯ |
| L(s) = 1 | − 4-s + 2/3·9-s + 3.61·11-s + 16-s − 25-s − 2/3·36-s − 3.61·44-s − 49-s − 64-s − 5/9·81-s + 2.41·99-s + 100-s + 7.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + 0.0760·173-s + 3.61·176-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.516743543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.516743543\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−9.565669527928134740582791013106, −9.403214286508681488348899166089, −8.796244736531070914431951008442, −8.505981583018836732726455405487, −7.75189914679495568519073220851, −7.15422930778059436718130427819, −6.59650246191371029736521081082, −6.23573005751784261986996663855, −5.62862706408842449384753513889, −4.71490216228862847097129025308, −4.22452857137103763944107380300, −3.85723965613467250065129788647, −3.34060628084415810888054474984, −1.76653615542448958592691323721, −1.13722799333171730144948342316,
1.13722799333171730144948342316, 1.76653615542448958592691323721, 3.34060628084415810888054474984, 3.85723965613467250065129788647, 4.22452857137103763944107380300, 4.71490216228862847097129025308, 5.62862706408842449384753513889, 6.23573005751784261986996663855, 6.59650246191371029736521081082, 7.15422930778059436718130427819, 7.75189914679495568519073220851, 8.505981583018836732726455405487, 8.796244736531070914431951008442, 9.403214286508681488348899166089, 9.565669527928134740582791013106
