L(s) = 1 | + 1.15·2-s − 3-s − 0.664·4-s + 5-s − 1.15·6-s + 3.45·7-s − 3.07·8-s + 9-s + 1.15·10-s + 3.68·11-s + 0.664·12-s + 13-s + 3.98·14-s − 15-s − 2.22·16-s + 5.55·17-s + 1.15·18-s + 7.37·19-s − 0.664·20-s − 3.45·21-s + 4.25·22-s − 2.65·23-s + 3.07·24-s + 25-s + 1.15·26-s − 27-s − 2.29·28-s + ⋯ |
L(s) = 1 | + 0.817·2-s − 0.577·3-s − 0.332·4-s + 0.447·5-s − 0.471·6-s + 1.30·7-s − 1.08·8-s + 0.333·9-s + 0.365·10-s + 1.11·11-s + 0.191·12-s + 0.277·13-s + 1.06·14-s − 0.258·15-s − 0.557·16-s + 1.34·17-s + 0.272·18-s + 1.69·19-s − 0.148·20-s − 0.753·21-s + 0.907·22-s − 0.554·23-s + 0.628·24-s + 0.200·25-s + 0.226·26-s − 0.192·27-s − 0.433·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.253885130\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.253885130\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
good | 2 | \( 1 - 1.15T + 2T^{2} \) |
| 7 | \( 1 - 3.45T + 7T^{2} \) |
| 11 | \( 1 - 3.68T + 11T^{2} \) |
| 17 | \( 1 - 5.55T + 17T^{2} \) |
| 19 | \( 1 - 7.37T + 19T^{2} \) |
| 23 | \( 1 + 2.65T + 23T^{2} \) |
| 29 | \( 1 + 7.40T + 29T^{2} \) |
| 37 | \( 1 - 6.10T + 37T^{2} \) |
| 41 | \( 1 - 3.82T + 41T^{2} \) |
| 43 | \( 1 - 4.86T + 43T^{2} \) |
| 47 | \( 1 + 7.44T + 47T^{2} \) |
| 53 | \( 1 + 5.58T + 53T^{2} \) |
| 59 | \( 1 + 5.32T + 59T^{2} \) |
| 61 | \( 1 - 0.0429T + 61T^{2} \) |
| 67 | \( 1 - 8.97T + 67T^{2} \) |
| 71 | \( 1 - 6.99T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 + 16.2T + 79T^{2} \) |
| 83 | \( 1 + 7.76T + 83T^{2} \) |
| 89 | \( 1 - 4.35T + 89T^{2} \) |
| 97 | \( 1 - 0.514T + 97T^{2} \) |
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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
−7.911491130870630838561957629855, −7.40047283104572891274341953580, −6.29596158534258495817448629654, −5.71590313315737598700801148050, −5.27541037524658687807259373477, −4.53261766395410290809320309503, −3.85026548555025878651152233581, −3.05354186245723757566765773438, −1.67044995360846724091084261448, −0.965894180599438061777675938086,
0.965894180599438061777675938086, 1.67044995360846724091084261448, 3.05354186245723757566765773438, 3.85026548555025878651152233581, 4.53261766395410290809320309503, 5.27541037524658687807259373477, 5.71590313315737598700801148050, 6.29596158534258495817448629654, 7.40047283104572891274341953580, 7.911491130870630838561957629855