| L(s) = 1 | − 9-s − 2·11-s − 4·16-s + 4·19-s + 4·29-s − 12·31-s − 10·41-s + 5·49-s − 16·59-s + 26·61-s − 10·71-s + 6·79-s + 81-s + 30·89-s + 2·99-s + 32·109-s − 19·121-s + 127-s + 131-s + 137-s + 139-s + 4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 0.603·11-s − 16-s + 0.917·19-s + 0.742·29-s − 2.15·31-s − 1.56·41-s + 5/7·49-s − 2.08·59-s + 3.32·61-s − 1.18·71-s + 0.675·79-s + 1/9·81-s + 3.17·89-s + 0.201·99-s + 3.06·109-s − 1.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.316805305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.316805305\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 95 T^{2} + p^{2} T^{4} \) |
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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
−10.31735317396824768809502703642, −9.706071431802248887959096682323, −9.422242296943292539054103396031, −8.850491829005862190419234371682, −8.735931152600397430769209815601, −8.113749870469077080885296620989, −7.70514096117492222865427021471, −7.11358636170508588123127764353, −7.09344020224328043004654971087, −6.29844590597326332851426982424, −5.98898085612776500076704849768, −5.32396479233902370931894440031, −5.02313578734529331417150860930, −4.67094723650139846474220326037, −3.74830190083547619016975775396, −3.55769438444294647887380150436, −2.81640120225307199592208901711, −2.25390446665196758907232158288, −1.64623953612891172505083644100, −0.52374012234064004903235829385,
0.52374012234064004903235829385, 1.64623953612891172505083644100, 2.25390446665196758907232158288, 2.81640120225307199592208901711, 3.55769438444294647887380150436, 3.74830190083547619016975775396, 4.67094723650139846474220326037, 5.02313578734529331417150860930, 5.32396479233902370931894440031, 5.98898085612776500076704849768, 6.29844590597326332851426982424, 7.09344020224328043004654971087, 7.11358636170508588123127764353, 7.70514096117492222865427021471, 8.113749870469077080885296620989, 8.735931152600397430769209815601, 8.850491829005862190419234371682, 9.422242296943292539054103396031, 9.706071431802248887959096682323, 10.31735317396824768809502703642
