| L(s) = 1 | + 27·3-s − 683·7-s + 729·9-s − 3.52e3·13-s + 1.28e4·19-s − 1.84e4·21-s + 1.96e4·27-s + 2.39e4·31-s + 8.92e4·37-s − 9.52e4·39-s + 1.53e5·43-s + 3.48e5·49-s + 3.46e5·57-s + 6.29e4·61-s − 4.97e5·63-s − 4.12e5·67-s − 6.38e5·73-s − 2.04e5·79-s + 5.31e5·81-s + 2.40e6·91-s + 6.46e5·93-s + 1.55e6·97-s − 1.12e6·103-s − 1.34e5·109-s + 2.40e6·111-s − 2.57e6·117-s + ⋯ |
| L(s) = 1 | + 3-s − 1.99·7-s + 9-s − 1.60·13-s + 1.87·19-s − 1.99·21-s + 27-s + 0.803·31-s + 1.76·37-s − 1.60·39-s + 1.93·43-s + 2.96·49-s + 1.87·57-s + 0.277·61-s − 1.99·63-s − 1.37·67-s − 1.64·73-s − 0.415·79-s + 81-s + 3.19·91-s + 0.803·93-s + 1.70·97-s − 1.03·103-s − 0.103·109-s + 1.76·111-s − 1.60·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.256440846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.256440846\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{3} T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 683 T + p^{6} T^{2} \) |
| 11 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 13 | \( 1 + 3527 T + p^{6} T^{2} \) |
| 17 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 19 | \( 1 - 12851 T + p^{6} T^{2} \) |
| 23 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 29 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 31 | \( 1 - 23939 T + p^{6} T^{2} \) |
| 37 | \( 1 - 89206 T + p^{6} T^{2} \) |
| 41 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 43 | \( 1 - 153973 T + p^{6} T^{2} \) |
| 47 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 53 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 59 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 61 | \( 1 - 62999 T + p^{6} T^{2} \) |
| 67 | \( 1 + 412523 T + p^{6} T^{2} \) |
| 71 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 73 | \( 1 + 638066 T + p^{6} T^{2} \) |
| 79 | \( 1 + 204622 T + p^{6} T^{2} \) |
| 83 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 89 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 97 | \( 1 - 1551817 T + p^{6} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.21891247848083594849306353937, −9.641377229889583538969747128026, −9.160536404848296932533572070511, −7.63780729358637688761584375564, −7.08119515606047213313302868885, −5.87836081608197881139223547800, −4.38805798742535981070531383215, −3.13421294108187759312052418983, −2.60194262976892909566011988310, −0.72798134064106747651451933812,
0.72798134064106747651451933812, 2.60194262976892909566011988310, 3.13421294108187759312052418983, 4.38805798742535981070531383215, 5.87836081608197881139223547800, 7.08119515606047213313302868885, 7.63780729358637688761584375564, 9.160536404848296932533572070511, 9.641377229889583538969747128026, 10.21891247848083594849306353937
