| L(s) = 1 | − 5-s + 2·7-s + 2·11-s − 13-s + 6·17-s − 4·19-s + 6·23-s + 5·25-s − 29-s − 8·31-s − 2·35-s + 2·37-s + 2·41-s + 10·43-s + 4·47-s + 7·49-s − 20·53-s − 2·55-s + 4·59-s − 9·61-s + 65-s − 14·67-s − 20·71-s − 18·73-s + 4·77-s + 10·79-s + 12·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s + 0.603·11-s − 0.277·13-s + 1.45·17-s − 0.917·19-s + 1.25·23-s + 25-s − 0.185·29-s − 1.43·31-s − 0.338·35-s + 0.328·37-s + 0.312·41-s + 1.52·43-s + 0.583·47-s + 49-s − 2.74·53-s − 0.269·55-s + 0.520·59-s − 1.15·61-s + 0.124·65-s − 1.71·67-s − 2.37·71-s − 2.10·73-s + 0.455·77-s + 1.12·79-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.471514174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.471514174\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 9 T + 20 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 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
−9.074219517910107170123169685523, −8.774174635496412521819271841852, −8.374157244671197229888976275219, −7.63095723746092022357266551524, −7.52936688303693474193662169505, −7.52153506196647021994677964460, −6.86089720500156528603793421175, −6.27498215342652458839939271201, −6.05295895014871378751704729591, −5.56990365191537811955551884258, −5.12159663933340461181337972183, −4.64008079958881044681754947750, −4.42455926541881354890315333849, −3.90391421929595213886614638042, −3.33540894067428484951595805272, −3.00388209189794220041371257209, −2.46556638195925482025849912338, −1.63510093334386925736397163060, −1.35091943175233921764231740144, −0.55197987167882812077438211262,
0.55197987167882812077438211262, 1.35091943175233921764231740144, 1.63510093334386925736397163060, 2.46556638195925482025849912338, 3.00388209189794220041371257209, 3.33540894067428484951595805272, 3.90391421929595213886614638042, 4.42455926541881354890315333849, 4.64008079958881044681754947750, 5.12159663933340461181337972183, 5.56990365191537811955551884258, 6.05295895014871378751704729591, 6.27498215342652458839939271201, 6.86089720500156528603793421175, 7.52153506196647021994677964460, 7.52936688303693474193662169505, 7.63095723746092022357266551524, 8.374157244671197229888976275219, 8.774174635496412521819271841852, 9.074219517910107170123169685523
