| L(s) = 1 | − 2·3-s + 2·5-s + 3·9-s − 12·13-s − 4·15-s − 25-s − 4·27-s + 16·31-s − 4·37-s + 24·39-s − 4·41-s + 8·43-s + 6·45-s + 10·49-s + 12·53-s − 24·65-s + 16·67-s + 24·71-s + 2·75-s + 5·81-s − 8·83-s − 20·89-s − 32·93-s + 24·107-s + 8·111-s − 36·117-s + 18·121-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 9-s − 3.32·13-s − 1.03·15-s − 1/5·25-s − 0.769·27-s + 2.87·31-s − 0.657·37-s + 3.84·39-s − 0.624·41-s + 1.21·43-s + 0.894·45-s + 10/7·49-s + 1.64·53-s − 2.97·65-s + 1.95·67-s + 2.84·71-s + 0.230·75-s + 5/9·81-s − 0.878·83-s − 2.11·89-s − 3.31·93-s + 2.32·107-s + 0.759·111-s − 3.32·117-s + 1.63·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.760781536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.760781536\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.553581912343736601123640740697, −8.320474709041013846904569692056, −7.87976864954381779992624312889, −7.37435437745216037539003831738, −7.00391833884896956190005850099, −7.00074252106946386729288262558, −6.44437738256167999445675951038, −6.06467204441022122435988759374, −5.54262494717769131249850345506, −5.37891102645110930637204927089, −4.92183341417785139541281227482, −4.76392071969394491282135993248, −4.16665499899804697515247924569, −3.93149578012733630907591592964, −2.85210407774094641046806171620, −2.75999272539017993770504633101, −2.07046516807725488064017094206, −1.99012172003255131090625225102, −0.809691506752474739841906517454, −0.56198214387489170079528983524,
0.56198214387489170079528983524, 0.809691506752474739841906517454, 1.99012172003255131090625225102, 2.07046516807725488064017094206, 2.75999272539017993770504633101, 2.85210407774094641046806171620, 3.93149578012733630907591592964, 4.16665499899804697515247924569, 4.76392071969394491282135993248, 4.92183341417785139541281227482, 5.37891102645110930637204927089, 5.54262494717769131249850345506, 6.06467204441022122435988759374, 6.44437738256167999445675951038, 7.00074252106946386729288262558, 7.00391833884896956190005850099, 7.37435437745216037539003831738, 7.87976864954381779992624312889, 8.320474709041013846904569692056, 8.553581912343736601123640740697
