| L(s) = 1 | − 3·3-s + 4·4-s + 11·7-s + 9·9-s − 12·12-s − 22·13-s + 16·16-s + 11·19-s − 33·21-s + 25·25-s − 27·27-s + 44·28-s + 59·31-s + 36·36-s + 47·37-s + 66·39-s − 22·43-s − 48·48-s + 72·49-s − 88·52-s − 33·57-s − 121·61-s + 99·63-s + 64·64-s − 13·67-s + 143·73-s − 75·75-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s + 11/7·7-s + 9-s − 12-s − 1.69·13-s + 16-s + 0.578·19-s − 1.57·21-s + 25-s − 27-s + 11/7·28-s + 1.90·31-s + 36-s + 1.27·37-s + 1.69·39-s − 0.511·43-s − 48-s + 1.46·49-s − 1.69·52-s − 0.578·57-s − 1.98·61-s + 11/7·63-s + 64-s − 0.194·67-s + 1.95·73-s − 75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.812812960\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.812812960\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p T \) |
| 11 | \( 1 \) |
| good | 2 | \( ( 1 - p T )( 1 + p T ) \) |
| 5 | \( ( 1 - p T )( 1 + p T ) \) |
| 7 | \( 1 - 11 T + p^{2} T^{2} \) |
| 13 | \( 1 + 22 T + p^{2} T^{2} \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( 1 - 11 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 - 59 T + p^{2} T^{2} \) |
| 37 | \( 1 - 47 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 + 22 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 + 121 T + p^{2} T^{2} \) |
| 67 | \( 1 + 13 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 - 143 T + p^{2} T^{2} \) |
| 79 | \( 1 - 11 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 + 169 T + p^{2} T^{2} \) |
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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
−11.30892749158712078603163940895, −10.56130346288221277108704863632, −9.695028722715903849503364508373, −8.034501373878848073185864004123, −7.39348566138595618180256575435, −6.44330728211004478990456938699, −5.22998109496337717304415858420, −4.59457931614714919593781088981, −2.54541508786044517333468954731, −1.21910497016182117701152442657,
1.21910497016182117701152442657, 2.54541508786044517333468954731, 4.59457931614714919593781088981, 5.22998109496337717304415858420, 6.44330728211004478990456938699, 7.39348566138595618180256575435, 8.034501373878848073185864004123, 9.695028722715903849503364508373, 10.56130346288221277108704863632, 11.30892749158712078603163940895
