| L(s) = 1 | − 3-s + 2·4-s + 5·7-s − 6·11-s − 2·12-s + 10·13-s + 6·17-s + 19-s − 5·21-s + 23-s + 5·25-s + 27-s + 10·28-s + 12·29-s − 5·31-s + 6·33-s + 7·37-s − 10·39-s − 2·43-s − 12·44-s − 6·47-s + 18·49-s − 6·51-s + 20·52-s − 12·53-s − 57-s + 6·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 4-s + 1.88·7-s − 1.80·11-s − 0.577·12-s + 2.77·13-s + 1.45·17-s + 0.229·19-s − 1.09·21-s + 0.208·23-s + 25-s + 0.192·27-s + 1.88·28-s + 2.22·29-s − 0.898·31-s + 1.04·33-s + 1.15·37-s − 1.60·39-s − 0.304·43-s − 1.80·44-s − 0.875·47-s + 18/7·49-s − 0.840·51-s + 2.77·52-s − 1.64·53-s − 0.132·57-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 233289 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 233289 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.586471487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.586471487\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| 23 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−11.04874881047402161388100752147, −10.87699742890481836393325049772, −10.59552810844395674561022520822, −10.24188879175280374370204475737, −9.473331339187304138948343335740, −8.666904667437368768231839564594, −8.310238398719040413340336024257, −8.215957292124443851724440526804, −7.51641584421071846187434081491, −7.28709521874879857386142411812, −6.29181313736809311815566810704, −6.23549363334136228540821825558, −5.53684978854764559278433097917, −5.14138898611262881020016876981, −4.67814758332579250034171209939, −3.98305005149303116711957671665, −2.93954750076820013256659025927, −2.81032073272351323418807827555, −1.41228084486232279061352243294, −1.33050985624522978346679624229,
1.33050985624522978346679624229, 1.41228084486232279061352243294, 2.81032073272351323418807827555, 2.93954750076820013256659025927, 3.98305005149303116711957671665, 4.67814758332579250034171209939, 5.14138898611262881020016876981, 5.53684978854764559278433097917, 6.23549363334136228540821825558, 6.29181313736809311815566810704, 7.28709521874879857386142411812, 7.51641584421071846187434081491, 8.215957292124443851724440526804, 8.310238398719040413340336024257, 8.666904667437368768231839564594, 9.473331339187304138948343335740, 10.24188879175280374370204475737, 10.59552810844395674561022520822, 10.87699742890481836393325049772, 11.04874881047402161388100752147
