L(s) = 1 | + 1.85·2-s − 2.11·3-s + 1.42·4-s + 5-s − 3.91·6-s − 4.74·7-s − 1.05·8-s + 1.46·9-s + 1.85·10-s − 11-s − 3.01·12-s + 3.70·13-s − 8.78·14-s − 2.11·15-s − 4.81·16-s − 5.22·17-s + 2.71·18-s − 4.97·19-s + 1.42·20-s + 10.0·21-s − 1.85·22-s + 5.98·23-s + 2.23·24-s + 25-s + 6.86·26-s + 3.24·27-s − 6.77·28-s + ⋯ |
L(s) = 1 | + 1.30·2-s − 1.22·3-s + 0.713·4-s + 0.447·5-s − 1.59·6-s − 1.79·7-s − 0.374·8-s + 0.488·9-s + 0.585·10-s − 0.301·11-s − 0.871·12-s + 1.02·13-s − 2.34·14-s − 0.545·15-s − 1.20·16-s − 1.26·17-s + 0.640·18-s − 1.14·19-s + 0.319·20-s + 2.18·21-s − 0.394·22-s + 1.24·23-s + 0.457·24-s + 0.200·25-s + 1.34·26-s + 0.623·27-s − 1.28·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.242403171\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.242403171\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
good | 2 | \( 1 - 1.85T + 2T^{2} \) |
| 3 | \( 1 + 2.11T + 3T^{2} \) |
| 7 | \( 1 + 4.74T + 7T^{2} \) |
| 13 | \( 1 - 3.70T + 13T^{2} \) |
| 17 | \( 1 + 5.22T + 17T^{2} \) |
| 19 | \( 1 + 4.97T + 19T^{2} \) |
| 23 | \( 1 - 5.98T + 23T^{2} \) |
| 29 | \( 1 + 1.29T + 29T^{2} \) |
| 31 | \( 1 + 1.51T + 31T^{2} \) |
| 37 | \( 1 - 0.300T + 37T^{2} \) |
| 41 | \( 1 - 8.90T + 41T^{2} \) |
| 43 | \( 1 + 1.23T + 43T^{2} \) |
| 47 | \( 1 + 4.65T + 47T^{2} \) |
| 53 | \( 1 - 1.98T + 53T^{2} \) |
| 59 | \( 1 - 5.51T + 59T^{2} \) |
| 61 | \( 1 - 0.0688T + 61T^{2} \) |
| 67 | \( 1 - 4.54T + 67T^{2} \) |
| 71 | \( 1 + 4.59T + 71T^{2} \) |
| 79 | \( 1 - 8.84T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 - 5.11T + 97T^{2} \) |
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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
−8.730953528345050232591491977024, −7.09078493993919336780419211425, −6.41429115774881995466621712972, −6.24108721881835197283148013972, −5.57093872161226439773259881707, −4.77502214834100533223886848952, −3.98833202567009565431957344739, −3.17718823474513944453512853537, −2.34247156037800412349951536298, −0.52408857258537604252559117373,
0.52408857258537604252559117373, 2.34247156037800412349951536298, 3.17718823474513944453512853537, 3.98833202567009565431957344739, 4.77502214834100533223886848952, 5.57093872161226439773259881707, 6.24108721881835197283148013972, 6.41429115774881995466621712972, 7.09078493993919336780419211425, 8.730953528345050232591491977024