| L(s) = 1 | + 4.84·2-s − 3·3-s + 15.5·4-s + 10.4·5-s − 14.5·6-s − 8.74·7-s + 36.3·8-s + 9·9-s + 50.4·10-s + 18.6·11-s − 46.5·12-s + 81.8·13-s − 42.3·14-s − 31.2·15-s + 52.4·16-s + 17.6·17-s + 43.6·18-s + 66.3·19-s + 161.·20-s + 26.2·21-s + 90.4·22-s − 147.·23-s − 109.·24-s − 16.7·25-s + 396.·26-s − 27·27-s − 135.·28-s + ⋯ |
| L(s) = 1 | + 1.71·2-s − 0.577·3-s + 1.93·4-s + 0.930·5-s − 0.989·6-s − 0.471·7-s + 1.60·8-s + 0.333·9-s + 1.59·10-s + 0.511·11-s − 1.11·12-s + 1.74·13-s − 0.808·14-s − 0.537·15-s + 0.818·16-s + 0.252·17-s + 0.571·18-s + 0.801·19-s + 1.80·20-s + 0.272·21-s + 0.876·22-s − 1.33·23-s − 0.928·24-s − 0.134·25-s + 2.99·26-s − 0.192·27-s − 0.914·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1011 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.486642724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.486642724\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 337 | \( 1 + 337T \) |
| good | 2 | \( 1 - 4.84T + 8T^{2} \) |
| 5 | \( 1 - 10.4T + 125T^{2} \) |
| 7 | \( 1 + 8.74T + 343T^{2} \) |
| 11 | \( 1 - 18.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 81.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 17.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 66.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 147.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 107.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 206.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 201.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 292.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 182.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 306.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 611.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 521.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 798.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 591.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 848.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 11.4T + 3.89e5T^{2} \) |
| 79 | \( 1 - 170.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 697.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 45.5T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.77e3T + 9.12e5T^{2} \) |
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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
−9.823009190094512212591898065231, −8.853113510413052528603850466212, −7.45387665569435056431300239964, −6.45386266520877992702952122000, −5.89027373075237148313069741767, −5.54771221326616715462687437047, −4.19005299219900323754731020211, −3.61388827964629278680399100830, −2.36161741565728731854258136966, −1.18652058646895822239820867487,
1.18652058646895822239820867487, 2.36161741565728731854258136966, 3.61388827964629278680399100830, 4.19005299219900323754731020211, 5.54771221326616715462687437047, 5.89027373075237148313069741767, 6.45386266520877992702952122000, 7.45387665569435056431300239964, 8.853113510413052528603850466212, 9.823009190094512212591898065231
