| L(s) = 1 | + 2·2-s + 2·3-s + 2·4-s + 4·6-s − 3·9-s + 4·12-s + 12·13-s − 4·16-s − 6·18-s + 24·26-s − 14·27-s − 16·31-s − 8·32-s − 6·36-s + 4·37-s + 24·39-s + 14·41-s − 8·43-s − 8·48-s + 10·49-s + 24·52-s − 8·53-s − 28·54-s − 32·62-s − 8·64-s − 6·67-s + 4·71-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 4-s + 1.63·6-s − 9-s + 1.15·12-s + 3.32·13-s − 16-s − 1.41·18-s + 4.70·26-s − 2.69·27-s − 2.87·31-s − 1.41·32-s − 36-s + 0.657·37-s + 3.84·39-s + 2.18·41-s − 1.21·43-s − 1.15·48-s + 10/7·49-s + 3.32·52-s − 1.09·53-s − 3.81·54-s − 4.06·62-s − 64-s − 0.733·67-s + 0.474·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.579886124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.579886124\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
| 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
−13.09149042712371155878063908164, −12.52574877004745161300305694143, −11.59831050319704897894374322610, −11.20879490196904099588130641318, −11.18912127803446529845833477808, −10.56668685954520975625995829211, −9.455979881866298956160714822643, −9.076258228608318098395863290254, −8.651273692220162019958801482689, −8.422270169196647035790482997003, −7.67629297974390958312445629125, −7.00410578573193468008559493674, −6.06983312997787725821990912336, −5.80570529969163636188501911080, −5.61206085840098872648814443259, −4.34177989757842361199735439160, −3.76929377233581991444359565792, −3.39991572985135430136585964330, −2.82275725919584111160698625505, −1.77719864173063494618556328250,
1.77719864173063494618556328250, 2.82275725919584111160698625505, 3.39991572985135430136585964330, 3.76929377233581991444359565792, 4.34177989757842361199735439160, 5.61206085840098872648814443259, 5.80570529969163636188501911080, 6.06983312997787725821990912336, 7.00410578573193468008559493674, 7.67629297974390958312445629125, 8.422270169196647035790482997003, 8.651273692220162019958801482689, 9.076258228608318098395863290254, 9.455979881866298956160714822643, 10.56668685954520975625995829211, 11.18912127803446529845833477808, 11.20879490196904099588130641318, 11.59831050319704897894374322610, 12.52574877004745161300305694143, 13.09149042712371155878063908164
