L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 4.47·7-s − 8-s + 9-s − 5.23·11-s − 12-s − 4.47·13-s − 4.47·14-s + 16-s + 4·17-s − 18-s + 5.70·19-s − 4.47·21-s + 5.23·22-s − 23-s + 24-s + 4.47·26-s − 27-s + 4.47·28-s − 4.47·29-s − 2.47·31-s − 32-s + 5.23·33-s − 4·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s + 1.69·7-s − 0.353·8-s + 0.333·9-s − 1.57·11-s − 0.288·12-s − 1.24·13-s − 1.19·14-s + 0.250·16-s + 0.970·17-s − 0.235·18-s + 1.30·19-s − 0.975·21-s + 1.11·22-s − 0.208·23-s + 0.204·24-s + 0.877·26-s − 0.192·27-s + 0.845·28-s − 0.830·29-s − 0.444·31-s − 0.176·32-s + 0.911·33-s − 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 - 4.47T + 7T^{2} \) |
| 11 | \( 1 + 5.23T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 5.70T + 19T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 + 2.47T + 31T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4.76T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 + 5.23T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 - 0.763T + 61T^{2} \) |
| 67 | \( 1 + 9.70T + 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 - 4.47T + 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + 0.472T + 97T^{2} \) |
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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
−7.940872700534058904550003472975, −7.63905764608586473399946470462, −7.15221369035361607506146900366, −5.71362967403371723583071790923, −5.22392005810875671879540815122, −4.75915400332134220466615277380, −3.31280507006915568183011268632, −2.22742130129801859869135830667, −1.37862073498099448356577790316, 0,
1.37862073498099448356577790316, 2.22742130129801859869135830667, 3.31280507006915568183011268632, 4.75915400332134220466615277380, 5.22392005810875671879540815122, 5.71362967403371723583071790923, 7.15221369035361607506146900366, 7.63905764608586473399946470462, 7.940872700534058904550003472975