L(s) = 1 | + 1.43·2-s − 3-s + 0.0715·4-s − 2.86·5-s − 1.43·6-s − 7-s − 2.77·8-s + 9-s − 4.12·10-s − 0.920·11-s − 0.0715·12-s − 1.43·14-s + 2.86·15-s − 4.13·16-s + 2.98·17-s + 1.43·18-s − 7.98·19-s − 0.205·20-s + 21-s − 1.32·22-s − 4.01·23-s + 2.77·24-s + 3.21·25-s − 27-s − 0.0715·28-s + 1.05·29-s + 4.12·30-s + ⋯ |
L(s) = 1 | + 1.01·2-s − 0.577·3-s + 0.0357·4-s − 1.28·5-s − 0.587·6-s − 0.377·7-s − 0.981·8-s + 0.333·9-s − 1.30·10-s − 0.277·11-s − 0.0206·12-s − 0.384·14-s + 0.740·15-s − 1.03·16-s + 0.723·17-s + 0.339·18-s − 1.83·19-s − 0.0458·20-s + 0.218·21-s − 0.282·22-s − 0.837·23-s + 0.566·24-s + 0.643·25-s − 0.192·27-s − 0.0135·28-s + 0.195·29-s + 0.753·30-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
\(L(1)\) |
\(\approx\) |
\(0.7966827545\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7966827545\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.43T + 2T^{2} \) |
| 5 | \( 1 + 2.86T + 5T^{2} \) |
| 11 | \( 1 + 0.920T + 11T^{2} \) |
| 17 | \( 1 - 2.98T + 17T^{2} \) |
| 19 | \( 1 + 7.98T + 19T^{2} \) |
| 23 | \( 1 + 4.01T + 23T^{2} \) |
| 29 | \( 1 - 1.05T + 29T^{2} \) |
| 31 | \( 1 + 3.15T + 31T^{2} \) |
| 37 | \( 1 + 1.61T + 37T^{2} \) |
| 41 | \( 1 + 6.54T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 4.63T + 47T^{2} \) |
| 53 | \( 1 - 6.68T + 53T^{2} \) |
| 59 | \( 1 - 9.98T + 59T^{2} \) |
| 61 | \( 1 + 8.95T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 + 9.09T + 71T^{2} \) |
| 73 | \( 1 - 7.52T + 73T^{2} \) |
| 79 | \( 1 - 6.24T + 79T^{2} \) |
| 83 | \( 1 - 9.66T + 83T^{2} \) |
| 89 | \( 1 - 6.52T + 89T^{2} \) |
| 97 | \( 1 + 0.561T + 97T^{2} \) |
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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
−8.419881259795323909503473578432, −7.79065100597167444147430488402, −6.86230850896749068472086397764, −6.18889828183491282766318963538, −5.45411805715395611287178806723, −4.63029115955807021685226345862, −3.96269277104842562124143725575, −3.48769843016616878610169313552, −2.29178855665367447169691805832, −0.44026045152808842644955927199,
0.44026045152808842644955927199, 2.29178855665367447169691805832, 3.48769843016616878610169313552, 3.96269277104842562124143725575, 4.63029115955807021685226345862, 5.45411805715395611287178806723, 6.18889828183491282766318963538, 6.86230850896749068472086397764, 7.79065100597167444147430488402, 8.419881259795323909503473578432