| L(s) = 1 | + 3-s + 4-s + 3·5-s + 9-s − 3·11-s + 12-s + 3·15-s + 16-s + 3·20-s + 3·25-s + 27-s − 5·31-s − 3·33-s + 36-s + 6·37-s − 3·44-s + 3·45-s + 15·47-s + 48-s − 49-s + 12·53-s − 9·55-s − 9·59-s + 3·60-s + 64-s + 11·67-s + 6·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s + 1.34·5-s + 1/3·9-s − 0.904·11-s + 0.288·12-s + 0.774·15-s + 1/4·16-s + 0.670·20-s + 3/5·25-s + 0.192·27-s − 0.898·31-s − 0.522·33-s + 1/6·36-s + 0.986·37-s − 0.452·44-s + 0.447·45-s + 2.18·47-s + 0.144·48-s − 1/7·49-s + 1.64·53-s − 1.21·55-s − 1.17·59-s + 0.387·60-s + 1/8·64-s + 1.34·67-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.148329232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.148329232\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 - T \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 11 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 49 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 76 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−7.83630960782459246702368670631, −7.34074300660921690196233882751, −6.92844794017124464205032331029, −6.58485731398685110674647680843, −5.92884474908554529794203628746, −5.65363961407897134755171858103, −5.39833829393300374531407245859, −4.78245459926447467756787177998, −4.14652215173605001481489612377, −3.74994183418434448169873274229, −2.94798995584602831193561527093, −2.61773835171478720144864162247, −2.15187360143282254705737186306, −1.67906652572808769094667554108, −0.792717012846335120098215303377,
0.792717012846335120098215303377, 1.67906652572808769094667554108, 2.15187360143282254705737186306, 2.61773835171478720144864162247, 2.94798995584602831193561527093, 3.74994183418434448169873274229, 4.14652215173605001481489612377, 4.78245459926447467756787177998, 5.39833829393300374531407245859, 5.65363961407897134755171858103, 5.92884474908554529794203628746, 6.58485731398685110674647680843, 6.92844794017124464205032331029, 7.34074300660921690196233882751, 7.83630960782459246702368670631
