L(s) = 1 | − 2·11-s − 4·16-s + 14·31-s + 16·41-s + 10·49-s + 10·59-s + 24·61-s + 6·71-s + 20·79-s + 30·89-s − 4·101-s − 20·109-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 8·176-s + 179-s + 181-s + ⋯ |
L(s) = 1 | − 0.603·11-s − 16-s + 2.51·31-s + 2.49·41-s + 10/7·49-s + 1.30·59-s + 3.07·61-s + 0.712·71-s + 2.25·79-s + 3.17·89-s − 0.398·101-s − 1.91·109-s + 3/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.603·176-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.684820430\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.684820430\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
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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
−9.165084479317677098385494198288, −8.660696378935712273189641518391, −8.328148512145031190969761258208, −8.067347954202489738830910533190, −7.58060360641750635872273022794, −7.26801250680856411924011612348, −6.74198575776120250887442349566, −6.46507675682819691012914834694, −6.12413043052292813165365685578, −5.48839480308999129741362162432, −5.29958517375224659287044402422, −4.59718540584982485388701259771, −4.53131206918408403275588185363, −3.80720158844220703298532645371, −3.58582445546653028719375067578, −2.67544941524919290821879443709, −2.39339510032273414591697618803, −2.20193489479546506633400668487, −0.955373789394485655072388047414, −0.69904508467170606839526776832,
0.69904508467170606839526776832, 0.955373789394485655072388047414, 2.20193489479546506633400668487, 2.39339510032273414591697618803, 2.67544941524919290821879443709, 3.58582445546653028719375067578, 3.80720158844220703298532645371, 4.53131206918408403275588185363, 4.59718540584982485388701259771, 5.29958517375224659287044402422, 5.48839480308999129741362162432, 6.12413043052292813165365685578, 6.46507675682819691012914834694, 6.74198575776120250887442349566, 7.26801250680856411924011612348, 7.58060360641750635872273022794, 8.067347954202489738830910533190, 8.328148512145031190969761258208, 8.660696378935712273189641518391, 9.165084479317677098385494198288