L(s) = 1 | + 2·5-s + 17·9-s + 48·13-s + 2·17-s − 47·25-s + 48·29-s − 98·37-s − 96·41-s + 34·45-s − 50·53-s − 2·61-s + 96·65-s + 190·73-s + 208·81-s + 4·85-s + 190·89-s + 288·97-s + 146·101-s − 142·109-s + 192·113-s + 816·117-s − 47·121-s − 146·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
L(s) = 1 | + 2/5·5-s + 17/9·9-s + 3.69·13-s + 2/17·17-s − 1.87·25-s + 1.65·29-s − 2.64·37-s − 2.34·41-s + 0.755·45-s − 0.943·53-s − 0.0327·61-s + 1.47·65-s + 2.60·73-s + 2.56·81-s + 4/85·85-s + 2.13·89-s + 2.96·97-s + 1.44·101-s − 1.30·109-s + 1.69·113-s + 6.97·117-s − 0.388·121-s − 1.16·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(5.420543173\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.420543173\) |
\(L(2)\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 - 17 T^{2} + p^{4} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 47 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 673 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 1009 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 241 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 49 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 48 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 3122 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 1393 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 25 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6673 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 4753 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 866 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 95 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 10801 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 8594 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 95 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 144 T + p^{2} T^{2} )^{2} \) |
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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
−9.406719406852110997081303780287, −8.988989986484869830239968770130, −8.540410323146462936999189577183, −8.468752889184557884717042750151, −7.75551583448930248518713441600, −7.64319149725720321384762697754, −6.79243482474875009519434986261, −6.51857873185918851434755613298, −6.36812905603996545269053181644, −5.96497737865809028549843827158, −5.16670520761857642507571168290, −5.04606263594184858383930194039, −4.29688048091104946226541468456, −3.81288014354592734396648854040, −3.41876164858198585331449951423, −3.41245638391770114190056805296, −1.93309475750444386976493939301, −1.82915816527752535301155794634, −1.27849246210382294034575121558, −0.69621390256156682547718037803,
0.69621390256156682547718037803, 1.27849246210382294034575121558, 1.82915816527752535301155794634, 1.93309475750444386976493939301, 3.41245638391770114190056805296, 3.41876164858198585331449951423, 3.81288014354592734396648854040, 4.29688048091104946226541468456, 5.04606263594184858383930194039, 5.16670520761857642507571168290, 5.96497737865809028549843827158, 6.36812905603996545269053181644, 6.51857873185918851434755613298, 6.79243482474875009519434986261, 7.64319149725720321384762697754, 7.75551583448930248518713441600, 8.468752889184557884717042750151, 8.540410323146462936999189577183, 8.988989986484869830239968770130, 9.406719406852110997081303780287