L(s) = 1 | + 1.80·2-s − 3-s + 1.24·4-s + 2.07·5-s − 1.80·6-s + 7-s − 1.35·8-s + 9-s + 3.74·10-s − 5.11·11-s − 1.24·12-s + 1.80·14-s − 2.07·15-s − 4.93·16-s − 0.185·17-s + 1.80·18-s + 0.591·19-s + 2.59·20-s − 21-s − 9.21·22-s − 2.78·23-s + 1.35·24-s − 0.681·25-s − 27-s + 1.24·28-s − 9.95·29-s − 3.74·30-s + ⋯ |
L(s) = 1 | + 1.27·2-s − 0.577·3-s + 0.623·4-s + 0.929·5-s − 0.735·6-s + 0.377·7-s − 0.479·8-s + 0.333·9-s + 1.18·10-s − 1.54·11-s − 0.359·12-s + 0.481·14-s − 0.536·15-s − 1.23·16-s − 0.0448·17-s + 0.424·18-s + 0.135·19-s + 0.579·20-s − 0.218·21-s − 1.96·22-s − 0.581·23-s + 0.276·24-s − 0.136·25-s − 0.192·27-s + 0.235·28-s − 1.84·29-s − 0.683·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 1.80T + 2T^{2} \) |
| 5 | \( 1 - 2.07T + 5T^{2} \) |
| 11 | \( 1 + 5.11T + 11T^{2} \) |
| 17 | \( 1 + 0.185T + 17T^{2} \) |
| 19 | \( 1 - 0.591T + 19T^{2} \) |
| 23 | \( 1 + 2.78T + 23T^{2} \) |
| 29 | \( 1 + 9.95T + 29T^{2} \) |
| 31 | \( 1 + 5.23T + 31T^{2} \) |
| 37 | \( 1 - 2.97T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 5.25T + 43T^{2} \) |
| 47 | \( 1 + 12.6T + 47T^{2} \) |
| 53 | \( 1 + 5.51T + 53T^{2} \) |
| 59 | \( 1 + 6.46T + 59T^{2} \) |
| 61 | \( 1 - 8.43T + 61T^{2} \) |
| 67 | \( 1 + 9.58T + 67T^{2} \) |
| 71 | \( 1 - 1.60T + 71T^{2} \) |
| 73 | \( 1 + 9.17T + 73T^{2} \) |
| 79 | \( 1 - 5.42T + 79T^{2} \) |
| 83 | \( 1 + 3.99T + 83T^{2} \) |
| 89 | \( 1 - 9.63T + 89T^{2} \) |
| 97 | \( 1 - 3.05T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.921985375557260773542869716665, −7.35745876532567208837979041058, −6.18900266318512371234613653050, −5.77135390990314071939749895400, −5.24427352553012675372014203520, −4.56927566105528981575150012415, −3.64592370007199641868014261137, −2.59642790780257195610302602397, −1.83108858439896778286888614901, 0,
1.83108858439896778286888614901, 2.59642790780257195610302602397, 3.64592370007199641868014261137, 4.56927566105528981575150012415, 5.24427352553012675372014203520, 5.77135390990314071939749895400, 6.18900266318512371234613653050, 7.35745876532567208837979041058, 7.921985375557260773542869716665