| L(s) = 1 | + 2·5-s − 6·9-s − 4·11-s + 4·16-s + 14·19-s + 5·25-s + 18·29-s − 28·31-s − 4·41-s − 12·45-s − 4·49-s − 8·55-s + 18·59-s + 14·61-s − 2·71-s + 2·79-s + 8·80-s + 9·81-s − 22·89-s + 28·95-s + 24·99-s − 30·101-s + 30·109-s − 34·121-s + 22·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 2·9-s − 1.20·11-s + 16-s + 3.21·19-s + 25-s + 3.34·29-s − 5.02·31-s − 0.624·41-s − 1.78·45-s − 4/7·49-s − 1.07·55-s + 2.34·59-s + 1.79·61-s − 0.237·71-s + 0.225·79-s + 0.894·80-s + 81-s − 2.33·89-s + 2.87·95-s + 2.41·99-s − 2.98·101-s + 2.87·109-s − 3.09·121-s + 1.96·125-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9325833630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9325833630\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| good | 2 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} ) \) |
| 3 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )( 1 + 23 T^{2} + p^{2} T^{4} ) \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 23 | $C_2^3$ | \( 1 + 10 T^{2} - 429 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 82 T^{2} + 4875 T^{4} + 82 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 58 T^{2} + 1155 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 14 T + 143 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \) |
| 59 | $C_2^2$ | \( ( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 34 T^{2} - 3333 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + T - 70 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )( 1 + 143 T^{2} + p^{2} T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 11 T + 32 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 + 158 T^{2} + 15555 T^{4} + 158 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.28406807025431810497662859055, −10.19169022711037439401446761241, −9.679195793232533285153980504497, −9.532039439246845229554027544587, −9.398209563299963851999341288282, −8.806519243418086288951730869509, −8.537086427225120139224986603788, −8.479769171669099165672302364040, −7.997997415445342114601443402482, −7.83893619798098288182257797829, −7.22009183458066532805278103219, −6.99686756585096186919380791256, −6.98308444319837001016377361472, −6.24988958210879402343157399272, −5.66588674590193468099192679184, −5.64913526436897972261373285785, −5.40320872327279133319137090296, −5.20363376167158838329305437717, −4.88992968633206983173756073625, −3.99573480617704466686026911245, −3.37630375747422206201856635784, −3.23549390921444222657453093382, −2.76886642020995138513698274223, −2.32251101184469232458412983922, −1.27418545886597354390154600307,
1.27418545886597354390154600307, 2.32251101184469232458412983922, 2.76886642020995138513698274223, 3.23549390921444222657453093382, 3.37630375747422206201856635784, 3.99573480617704466686026911245, 4.88992968633206983173756073625, 5.20363376167158838329305437717, 5.40320872327279133319137090296, 5.64913526436897972261373285785, 5.66588674590193468099192679184, 6.24988958210879402343157399272, 6.98308444319837001016377361472, 6.99686756585096186919380791256, 7.22009183458066532805278103219, 7.83893619798098288182257797829, 7.997997415445342114601443402482, 8.479769171669099165672302364040, 8.537086427225120139224986603788, 8.806519243418086288951730869509, 9.398209563299963851999341288282, 9.532039439246845229554027544587, 9.679195793232533285153980504497, 10.19169022711037439401446761241, 10.28406807025431810497662859055
