L(s) = 1 | − 2·4-s + 5-s − 7-s + 11-s − 4·13-s + 4·16-s − 2·17-s − 6·19-s − 2·20-s + 5·23-s − 4·25-s + 2·28-s − 10·29-s + 31-s − 35-s − 5·37-s + 2·41-s − 8·43-s − 2·44-s − 8·47-s + 49-s + 8·52-s + 6·53-s + 55-s − 3·59-s − 2·61-s − 8·64-s + ⋯ |
L(s) = 1 | − 4-s + 0.447·5-s − 0.377·7-s + 0.301·11-s − 1.10·13-s + 16-s − 0.485·17-s − 1.37·19-s − 0.447·20-s + 1.04·23-s − 4/5·25-s + 0.377·28-s − 1.85·29-s + 0.179·31-s − 0.169·35-s − 0.821·37-s + 0.312·41-s − 1.21·43-s − 0.301·44-s − 1.16·47-s + 1/7·49-s + 1.10·52-s + 0.824·53-s + 0.134·55-s − 0.390·59-s − 0.256·61-s − 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−9.843866131006304498608069072697, −9.263232139707367468574678849511, −8.513651119993219671602342798866, −7.41775134048211629573821110448, −6.42370236549041572726553632076, −5.38913047752314397636287568574, −4.53418395180193611114078817717, −3.48522622489004363314719629746, −2.00735365908684561839702035924, 0,
2.00735365908684561839702035924, 3.48522622489004363314719629746, 4.53418395180193611114078817717, 5.38913047752314397636287568574, 6.42370236549041572726553632076, 7.41775134048211629573821110448, 8.513651119993219671602342798866, 9.263232139707367468574678849511, 9.843866131006304498608069072697