| L(s) = 1 | + 2·3-s + 3·9-s − 4·19-s + 7·25-s + 4·27-s − 18·29-s − 2·31-s − 4·37-s + 18·53-s − 8·57-s + 6·59-s + 14·75-s + 5·81-s + 30·83-s − 36·87-s − 4·93-s − 8·103-s − 8·109-s − 8·111-s − 12·113-s − 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s − 0.917·19-s + 7/5·25-s + 0.769·27-s − 3.34·29-s − 0.359·31-s − 0.657·37-s + 2.47·53-s − 1.05·57-s + 0.781·59-s + 1.61·75-s + 5/9·81-s + 3.29·83-s − 3.85·87-s − 0.414·93-s − 0.788·103-s − 0.766·109-s − 0.759·111-s − 1.12·113-s − 0.454·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.146482985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.146482985\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 155 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 119 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.021969942521490074614525745408, −8.819208174125599229354587656741, −8.622971019739054333990298511950, −7.962262603764789126290965356304, −7.61447029283636806153977649183, −7.45291502206463489730595841104, −6.92115446833376555040757999752, −6.56984864773079163300538120091, −6.20633624920199868461734217867, −5.38803813851764505679459487226, −5.34634570947769757537990730811, −4.85689434590501233970516167507, −3.95388238717926402880942663635, −3.92849097981636598163748821257, −3.63588145971488819911007379759, −2.83232988881421012280967612273, −2.48811849411808031473204208142, −1.94137547141932841602340435805, −1.51893147003307601688768426475, −0.54503241457184963027589456546,
0.54503241457184963027589456546, 1.51893147003307601688768426475, 1.94137547141932841602340435805, 2.48811849411808031473204208142, 2.83232988881421012280967612273, 3.63588145971488819911007379759, 3.92849097981636598163748821257, 3.95388238717926402880942663635, 4.85689434590501233970516167507, 5.34634570947769757537990730811, 5.38803813851764505679459487226, 6.20633624920199868461734217867, 6.56984864773079163300538120091, 6.92115446833376555040757999752, 7.45291502206463489730595841104, 7.61447029283636806153977649183, 7.962262603764789126290965356304, 8.622971019739054333990298511950, 8.819208174125599229354587656741, 9.021969942521490074614525745408
