L(s) = 1 | + 2·3-s − 4-s + 3·9-s − 2·12-s − 3·16-s + 12·23-s + 4·27-s + 8·31-s − 3·36-s + 22·37-s − 6·48-s − 2·49-s + 18·53-s − 12·59-s + 7·64-s + 4·67-s + 24·69-s − 12·71-s + 5·81-s + 18·89-s − 12·92-s + 16·93-s + 14·97-s − 28·103-s − 4·108-s + 44·111-s + 42·113-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 1/2·4-s + 9-s − 0.577·12-s − 3/4·16-s + 2.50·23-s + 0.769·27-s + 1.43·31-s − 1/2·36-s + 3.61·37-s − 0.866·48-s − 2/7·49-s + 2.47·53-s − 1.56·59-s + 7/8·64-s + 0.488·67-s + 2.88·69-s − 1.42·71-s + 5/9·81-s + 1.90·89-s − 1.25·92-s + 1.65·93-s + 1.42·97-s − 2.75·103-s − 0.384·108-s + 4.17·111-s + 3.95·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82355625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82355625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.206157536\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.206157536\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 11 | | \( 1 \) |
good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 79 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
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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
−7.84605671860960842263485631020, −7.56626177808390425503802566062, −7.43292872227637892959162504220, −6.77263142478702577736009670529, −6.74544508007790591047393054058, −6.31257080390574645679109099904, −5.69709533654698555277739253123, −5.65654718699107272828865157544, −4.82935284143458366473936122396, −4.71837506330001720979415272207, −4.34908154766420695757309958265, −4.25481102489424678687943795489, −3.45900852155314022953775734056, −3.26783080166911657500046920693, −2.83839711710534354804617616625, −2.36809759390906627924853916185, −2.30672533754142861420352165627, −1.43276252738858661342632756536, −0.898254053855667962944737968862, −0.67152987784046778169232541463,
0.67152987784046778169232541463, 0.898254053855667962944737968862, 1.43276252738858661342632756536, 2.30672533754142861420352165627, 2.36809759390906627924853916185, 2.83839711710534354804617616625, 3.26783080166911657500046920693, 3.45900852155314022953775734056, 4.25481102489424678687943795489, 4.34908154766420695757309958265, 4.71837506330001720979415272207, 4.82935284143458366473936122396, 5.65654718699107272828865157544, 5.69709533654698555277739253123, 6.31257080390574645679109099904, 6.74544508007790591047393054058, 6.77263142478702577736009670529, 7.43292872227637892959162504220, 7.56626177808390425503802566062, 7.84605671860960842263485631020