| L(s) = 1 | − 3·9-s + 6·29-s + 6·31-s + 6·41-s + 10·49-s + 16·59-s − 8·61-s + 14·71-s + 4·89-s + 36·101-s + 36·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 25·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 9-s + 1.11·29-s + 1.07·31-s + 0.937·41-s + 10/7·49-s + 2.08·59-s − 1.02·61-s + 1.66·71-s + 0.423·89-s + 3.58·101-s + 3.44·109-s − 2·121-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 + 1.92·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.997357287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.997357287\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 23 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−8.662486920376952591533446037602, −8.149097666441831610179014039050, −7.85205156031396279168738253681, −7.45314201601922274236887331474, −7.02404820238043737778499722383, −6.70273101460077679967451138801, −6.22224997531827549575503713729, −6.04178741119857415136783117697, −5.51064474057715498178683223258, −5.34321068353149304637587292023, −4.61060177125585013665299574561, −4.57194838248218024187446697513, −4.01931324585700801304247440971, −3.34227735495195328919822928698, −3.27879554403493564280515281657, −2.49210468537196315234766297334, −2.40255080554156027422688666389, −1.76183632493648659148647100219, −0.76528491901308749605670720691, −0.69843220659036803517831161922,
0.69843220659036803517831161922, 0.76528491901308749605670720691, 1.76183632493648659148647100219, 2.40255080554156027422688666389, 2.49210468537196315234766297334, 3.27879554403493564280515281657, 3.34227735495195328919822928698, 4.01931324585700801304247440971, 4.57194838248218024187446697513, 4.61060177125585013665299574561, 5.34321068353149304637587292023, 5.51064474057715498178683223258, 6.04178741119857415136783117697, 6.22224997531827549575503713729, 6.70273101460077679967451138801, 7.02404820238043737778499722383, 7.45314201601922274236887331474, 7.85205156031396279168738253681, 8.149097666441831610179014039050, 8.662486920376952591533446037602
