| L(s) = 1 | − 2.95·2-s + 0.709·4-s − 16.6·5-s − 30.2·7-s + 21.5·8-s + 49.0·10-s − 7.82·11-s + 48.9·13-s + 89.2·14-s − 69.1·16-s + 47.2·17-s + 48.8·19-s − 11.8·20-s + 23.1·22-s − 130.·23-s + 151.·25-s − 144.·26-s − 21.4·28-s + 145.·29-s − 122.·31-s + 32.0·32-s − 139.·34-s + 503.·35-s − 31.4·37-s − 144.·38-s − 357.·40-s − 123.·41-s + ⋯ |
| L(s) = 1 | − 1.04·2-s + 0.0886·4-s − 1.48·5-s − 1.63·7-s + 0.950·8-s + 1.55·10-s − 0.214·11-s + 1.04·13-s + 1.70·14-s − 1.08·16-s + 0.673·17-s + 0.589·19-s − 0.131·20-s + 0.223·22-s − 1.18·23-s + 1.21·25-s − 1.08·26-s − 0.144·28-s + 0.932·29-s − 0.708·31-s + 0.176·32-s − 0.702·34-s + 2.42·35-s − 0.139·37-s − 0.615·38-s − 1.41·40-s − 0.470·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1605799757\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1605799757\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 + 2.95T + 8T^{2} \) |
| 5 | \( 1 + 16.6T + 125T^{2} \) |
| 7 | \( 1 + 30.2T + 343T^{2} \) |
| 11 | \( 1 + 7.82T + 1.33e3T^{2} \) |
| 13 | \( 1 - 48.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 47.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 48.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 130.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 145.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 122.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 31.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + 123.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 306.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 143.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 285.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 260.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 604.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 517.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 246.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 882.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 34.8T + 5.71e5T^{2} \) |
| 89 | \( 1 - 810.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 192.T + 9.12e5T^{2} \) |
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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
−8.688443265837766809492389510418, −8.049852736526316738638574268600, −7.42046716394474560668812074575, −6.67281330806489233692634197347, −5.74179566295157911166022504931, −4.45842422259734005562997122305, −3.66327271188467637077124652753, −3.07096925649549873009262186514, −1.34160094242162130232963130956, −0.23352445526914772995484677290,
0.23352445526914772995484677290, 1.34160094242162130232963130956, 3.07096925649549873009262186514, 3.66327271188467637077124652753, 4.45842422259734005562997122305, 5.74179566295157911166022504931, 6.67281330806489233692634197347, 7.42046716394474560668812074575, 8.049852736526316738638574268600, 8.688443265837766809492389510418
