| L(s) = 1 | + 6·11-s + 10·13-s − 18·23-s + 7·25-s − 4·37-s − 6·47-s + 11·49-s − 6·59-s + 2·61-s − 24·71-s − 4·73-s + 30·83-s − 10·97-s + 24·107-s + 28·109-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 60·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 49·169-s + ⋯ |
| L(s) = 1 | + 1.80·11-s + 2.77·13-s − 3.75·23-s + 7/5·25-s − 0.657·37-s − 0.875·47-s + 11/7·49-s − 0.781·59-s + 0.256·61-s − 2.84·71-s − 0.468·73-s + 3.29·83-s − 1.01·97-s + 2.32·107-s + 2.68·109-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.01·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.76·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.525574198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.525574198\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 83 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
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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
−8.393526237327487574312389847081, −8.185629871350790923264469682347, −7.69882562424752996939984522489, −7.34272692619234445464609025590, −6.84852514401317058996626829784, −6.46650795648470891501974866485, −6.14820361739179933545649849314, −5.91842293930052484710406829764, −5.89799394227023465657228097171, −5.10580475496737430409185063734, −4.45168897462150204229962966188, −4.36479880481143878349669933340, −3.74967416156838715988115292678, −3.58574951045611044379978100732, −3.42086456742999963095921780757, −2.57055731503729850114646659205, −1.96009502243805928419317129873, −1.55561083441695293420056958594, −1.23484123063904837328669811932, −0.53259920371438026012653429510,
0.53259920371438026012653429510, 1.23484123063904837328669811932, 1.55561083441695293420056958594, 1.96009502243805928419317129873, 2.57055731503729850114646659205, 3.42086456742999963095921780757, 3.58574951045611044379978100732, 3.74967416156838715988115292678, 4.36479880481143878349669933340, 4.45168897462150204229962966188, 5.10580475496737430409185063734, 5.89799394227023465657228097171, 5.91842293930052484710406829764, 6.14820361739179933545649849314, 6.46650795648470891501974866485, 6.84852514401317058996626829784, 7.34272692619234445464609025590, 7.69882562424752996939984522489, 8.185629871350790923264469682347, 8.393526237327487574312389847081
