| L(s) = 1 | − 2·5-s + 2·9-s − 8·11-s − 2·19-s − 25-s − 4·29-s + 16·31-s − 12·41-s − 4·45-s + 10·49-s + 16·55-s − 8·59-s + 12·61-s − 32·71-s + 16·79-s − 5·81-s + 20·89-s + 4·95-s − 16·99-s + 12·101-s − 4·109-s + 26·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 2/3·9-s − 2.41·11-s − 0.458·19-s − 1/5·25-s − 0.742·29-s + 2.87·31-s − 1.87·41-s − 0.596·45-s + 10/7·49-s + 2.15·55-s − 1.04·59-s + 1.53·61-s − 3.79·71-s + 1.80·79-s − 5/9·81-s + 2.11·89-s + 0.410·95-s − 1.60·99-s + 1.19·101-s − 0.383·109-s + 2.36·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7692794429\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7692794429\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 178 T^{2} + p^{2} T^{4} \) |
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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
−9.880782130745058808654517468042, −9.200774305934452121600986859692, −8.778707145911594731614301289917, −8.313884147095107406321305557223, −7.930612704967571769842427199455, −7.891301947340583955684355685581, −7.29263476837926517814888242162, −7.00092467746203929968661592773, −6.50275635170095270811662087602, −5.84351947352599318538626794631, −5.65265511249086114796470726652, −4.93260382947236744360248063708, −4.57385193052375166052643085086, −4.44448359996504954205454824091, −3.50437497519300771871528427078, −3.33865801319378317892570592342, −2.44349689500075743121062098094, −2.35559704477697595954339574485, −1.31385920126656720218383332207, −0.35771347266546139396829529225,
0.35771347266546139396829529225, 1.31385920126656720218383332207, 2.35559704477697595954339574485, 2.44349689500075743121062098094, 3.33865801319378317892570592342, 3.50437497519300771871528427078, 4.44448359996504954205454824091, 4.57385193052375166052643085086, 4.93260382947236744360248063708, 5.65265511249086114796470726652, 5.84351947352599318538626794631, 6.50275635170095270811662087602, 7.00092467746203929968661592773, 7.29263476837926517814888242162, 7.891301947340583955684355685581, 7.930612704967571769842427199455, 8.313884147095107406321305557223, 8.778707145911594731614301289917, 9.200774305934452121600986859692, 9.880782130745058808654517468042
