| L(s) = 1 | − 4-s − 4·5-s + 6·9-s + 16-s + 12·19-s + 4·20-s + 11·25-s − 8·31-s − 6·36-s − 20·41-s − 24·45-s + 10·49-s + 12·59-s − 64-s − 12·76-s − 8·79-s − 4·80-s + 27·81-s − 12·89-s − 48·95-s − 11·100-s − 20·101-s + 40·109-s − 22·121-s + 8·124-s − 24·125-s + 127-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.78·5-s + 2·9-s + 1/4·16-s + 2.75·19-s + 0.894·20-s + 11/5·25-s − 1.43·31-s − 36-s − 3.12·41-s − 3.57·45-s + 10/7·49-s + 1.56·59-s − 1/8·64-s − 1.37·76-s − 0.900·79-s − 0.447·80-s + 3·81-s − 1.27·89-s − 4.92·95-s − 1.09·100-s − 1.99·101-s + 3.83·109-s − 2·121-s + 0.718·124-s − 2.14·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.183770704\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.183770704\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 37 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−11.59081892644473436960668470110, −11.40506968222612717889534197365, −10.62908174788076414305464825761, −10.25113719858917368291542275203, −9.783773818615702590309723032008, −9.438090247624500579410071547213, −8.798930465226186592441211260912, −8.322965245049030687385456798168, −7.79721975732335667065088838795, −7.35568832889714045281533432589, −7.01158820365425114598423808349, −6.84419121247274177434054961507, −5.46016000559177642959206737587, −5.30465217903894502846767683368, −4.49338082846149928806726260801, −4.17325041782622047298026959064, −3.40786179572380380383719684032, −3.30426030742188220577888812198, −1.71098677704674363115611356244, −0.820739007894080673431905223777,
0.820739007894080673431905223777, 1.71098677704674363115611356244, 3.30426030742188220577888812198, 3.40786179572380380383719684032, 4.17325041782622047298026959064, 4.49338082846149928806726260801, 5.30465217903894502846767683368, 5.46016000559177642959206737587, 6.84419121247274177434054961507, 7.01158820365425114598423808349, 7.35568832889714045281533432589, 7.79721975732335667065088838795, 8.322965245049030687385456798168, 8.798930465226186592441211260912, 9.438090247624500579410071547213, 9.783773818615702590309723032008, 10.25113719858917368291542275203, 10.62908174788076414305464825761, 11.40506968222612717889534197365, 11.59081892644473436960668470110
