| L(s) = 1 | + 5-s + 3·7-s + 5·11-s − 5·13-s + 2·17-s + 4·19-s − 23-s − 4·25-s + 9·29-s + 31-s + 3·35-s − 6·37-s − 3·41-s − 43-s − 3·47-s + 2·49-s − 2·53-s + 5·55-s + 11·59-s + 7·61-s − 5·65-s + 67-s + 4·71-s − 2·73-s + 15·77-s − 79-s + 83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.13·7-s + 1.50·11-s − 1.38·13-s + 0.485·17-s + 0.917·19-s − 0.208·23-s − 4/5·25-s + 1.67·29-s + 0.179·31-s + 0.507·35-s − 0.986·37-s − 0.468·41-s − 0.152·43-s − 0.437·47-s + 2/7·49-s − 0.274·53-s + 0.674·55-s + 1.43·59-s + 0.896·61-s − 0.620·65-s + 0.122·67-s + 0.474·71-s − 0.234·73-s + 1.70·77-s − 0.112·79-s + 0.109·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.178891095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.178891095\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−9.786481556435192006264902976817, −8.852175048728106831755803663857, −8.048167203366270559039286493432, −7.21159937416040143921402129373, −6.40427520216457499350445059134, −5.30217691197637718492882803293, −4.68851025585393844321739435382, −3.57334901540014761589778098736, −2.23509803185341693599086154104, −1.21735235993044136894134834772,
1.21735235993044136894134834772, 2.23509803185341693599086154104, 3.57334901540014761589778098736, 4.68851025585393844321739435382, 5.30217691197637718492882803293, 6.40427520216457499350445059134, 7.21159937416040143921402129373, 8.048167203366270559039286493432, 8.852175048728106831755803663857, 9.786481556435192006264902976817
