| L(s) = 1 | − 1.77·2-s + 1.13·4-s − 0.437·7-s + 1.52·8-s − 5.73·11-s + 13-s + 0.775·14-s − 4.98·16-s − 3.98·17-s + 4.77·19-s + 10.1·22-s − 0.337·23-s − 1.77·26-s − 0.498·28-s − 1.72·29-s − 7.86·31-s + 5.77·32-s + 7.05·34-s + 1.66·37-s − 8.45·38-s − 2.68·41-s + 2.27·43-s − 6.52·44-s + 0.598·46-s + 11.7·47-s − 6.80·49-s + 1.13·52-s + ⋯ |
| L(s) = 1 | − 1.25·2-s + 0.569·4-s − 0.165·7-s + 0.539·8-s − 1.72·11-s + 0.277·13-s + 0.207·14-s − 1.24·16-s − 0.965·17-s + 1.09·19-s + 2.16·22-s − 0.0704·23-s − 0.347·26-s − 0.0941·28-s − 0.319·29-s − 1.41·31-s + 1.02·32-s + 1.20·34-s + 0.273·37-s − 1.37·38-s − 0.419·41-s + 0.347·43-s − 0.984·44-s + 0.0882·46-s + 1.70·47-s − 0.972·49-s + 0.157·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5444058736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5444058736\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 1.77T + 2T^{2} \) |
| 7 | \( 1 + 0.437T + 7T^{2} \) |
| 11 | \( 1 + 5.73T + 11T^{2} \) |
| 17 | \( 1 + 3.98T + 17T^{2} \) |
| 19 | \( 1 - 4.77T + 19T^{2} \) |
| 23 | \( 1 + 0.337T + 23T^{2} \) |
| 29 | \( 1 + 1.72T + 29T^{2} \) |
| 31 | \( 1 + 7.86T + 31T^{2} \) |
| 37 | \( 1 - 1.66T + 37T^{2} \) |
| 41 | \( 1 + 2.68T + 41T^{2} \) |
| 43 | \( 1 - 2.27T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 - 14.2T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 + 2.25T + 61T^{2} \) |
| 67 | \( 1 - 2.17T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 2.55T + 73T^{2} \) |
| 79 | \( 1 - 2.55T + 79T^{2} \) |
| 83 | \( 1 - 5.41T + 83T^{2} \) |
| 89 | \( 1 - 4.72T + 89T^{2} \) |
| 97 | \( 1 + 14.7T + 97T^{2} \) |
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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
−8.845559551670141186454111226393, −8.011823397444313586726551600376, −7.52527174058329604805505049257, −6.85202466645579497323586117426, −5.67603517614060662780616236544, −5.02559152442015011765115296574, −3.97579368212590363559838141430, −2.78244430481838812172564659737, −1.89216588988696207664075349000, −0.53344791111141610447818392208,
0.53344791111141610447818392208, 1.89216588988696207664075349000, 2.78244430481838812172564659737, 3.97579368212590363559838141430, 5.02559152442015011765115296574, 5.67603517614060662780616236544, 6.85202466645579497323586117426, 7.52527174058329604805505049257, 8.011823397444313586726551600376, 8.845559551670141186454111226393
