L(s) = 1 | − 0.339·2-s − 1.88·4-s + 5-s − 0.660·7-s + 1.32·8-s − 0.339·10-s + 0.679·11-s + 0.224·14-s + 3.32·16-s − 7.42·17-s − 0.115·19-s − 1.88·20-s − 0.231·22-s + 7.76·23-s + 25-s + 1.24·28-s − 5.54·29-s + 9.97·31-s − 3.76·32-s + 2.52·34-s − 0.660·35-s − 9.76·37-s + 0.0392·38-s + 1.32·40-s + 4.22·41-s − 0.544·43-s − 1.28·44-s + ⋯ |
L(s) = 1 | − 0.240·2-s − 0.942·4-s + 0.447·5-s − 0.249·7-s + 0.466·8-s − 0.107·10-s + 0.204·11-s + 0.0599·14-s + 0.830·16-s − 1.80·17-s − 0.0265·19-s − 0.421·20-s − 0.0492·22-s + 1.61·23-s + 0.200·25-s + 0.235·28-s − 1.02·29-s + 1.79·31-s − 0.666·32-s + 0.433·34-s − 0.111·35-s − 1.60·37-s + 0.00636·38-s + 0.208·40-s + 0.659·41-s − 0.0830·43-s − 0.193·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 0.339T + 2T^{2} \) |
| 7 | \( 1 + 0.660T + 7T^{2} \) |
| 11 | \( 1 - 0.679T + 11T^{2} \) |
| 17 | \( 1 + 7.42T + 17T^{2} \) |
| 19 | \( 1 + 0.115T + 19T^{2} \) |
| 23 | \( 1 - 7.76T + 23T^{2} \) |
| 29 | \( 1 + 5.54T + 29T^{2} \) |
| 31 | \( 1 - 9.97T + 31T^{2} \) |
| 37 | \( 1 + 9.76T + 37T^{2} \) |
| 41 | \( 1 - 4.22T + 41T^{2} \) |
| 43 | \( 1 + 0.544T + 43T^{2} \) |
| 47 | \( 1 + 5.01T + 47T^{2} \) |
| 53 | \( 1 + 0.679T + 53T^{2} \) |
| 59 | \( 1 - 2.22T + 59T^{2} \) |
| 61 | \( 1 + 4.20T + 61T^{2} \) |
| 67 | \( 1 - 7.63T + 67T^{2} \) |
| 71 | \( 1 - 7.31T + 71T^{2} \) |
| 73 | \( 1 + 8.01T + 73T^{2} \) |
| 79 | \( 1 - 9.97T + 79T^{2} \) |
| 83 | \( 1 - 1.76T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 + 9.90T + 97T^{2} \) |
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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
−7.58867237689709544653542528707, −6.72448111164943452419810166337, −6.31339064940290526172002092441, −5.18726328580393753610357782212, −4.82976939833189375859134873866, −3.99627831995318337721132492919, −3.16371099255121964907188189966, −2.17749890594373206039450747732, −1.14176125531208770803233411891, 0,
1.14176125531208770803233411891, 2.17749890594373206039450747732, 3.16371099255121964907188189966, 3.99627831995318337721132492919, 4.82976939833189375859134873866, 5.18726328580393753610357782212, 6.31339064940290526172002092441, 6.72448111164943452419810166337, 7.58867237689709544653542528707