| L(s) = 1 | − 0.481·2-s + 3·3-s − 7.76·4-s + 20.6·5-s − 1.44·6-s − 14.3·7-s + 7.59·8-s + 9·9-s − 9.96·10-s + 22.4·11-s − 23.3·12-s + 40.4·13-s + 6.91·14-s + 62.0·15-s + 58.4·16-s − 70.2·17-s − 4.33·18-s + 116.·19-s − 160.·20-s − 43.0·21-s − 10.8·22-s + 23·23-s + 22.7·24-s + 302.·25-s − 19.4·26-s + 27·27-s + 111.·28-s + ⋯ |
| L(s) = 1 | − 0.170·2-s + 0.577·3-s − 0.970·4-s + 1.84·5-s − 0.0983·6-s − 0.775·7-s + 0.335·8-s + 0.333·9-s − 0.315·10-s + 0.615·11-s − 0.560·12-s + 0.862·13-s + 0.132·14-s + 1.06·15-s + 0.913·16-s − 1.00·17-s − 0.0567·18-s + 1.40·19-s − 1.79·20-s − 0.447·21-s − 0.104·22-s + 0.208·23-s + 0.193·24-s + 2.41·25-s − 0.147·26-s + 0.192·27-s + 0.752·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.179011044\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.179011044\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 23 | \( 1 - 23T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 + 0.481T + 8T^{2} \) |
| 5 | \( 1 - 20.6T + 125T^{2} \) |
| 7 | \( 1 + 14.3T + 343T^{2} \) |
| 11 | \( 1 - 22.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 40.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 70.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 116.T + 6.85e3T^{2} \) |
| 31 | \( 1 - 93.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 82.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 168.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 383.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 253.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 574.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 677.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 886.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 665.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 262.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.03e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 172.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 748.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 565.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.59e3T + 9.12e5T^{2} \) |
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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
−8.922469655433439173530821780472, −8.458022874488926578659876600877, −7.11942832489427294727730656240, −6.38553623705013645844052270144, −5.65682108264632978896112446832, −4.83135146255233257523239144594, −3.73317537288885717794347087957, −2.88813724879474032790649206687, −1.73938776777820488940906233823, −0.881115068594417926401640917737,
0.881115068594417926401640917737, 1.73938776777820488940906233823, 2.88813724879474032790649206687, 3.73317537288885717794347087957, 4.83135146255233257523239144594, 5.65682108264632978896112446832, 6.38553623705013645844052270144, 7.11942832489427294727730656240, 8.458022874488926578659876600877, 8.922469655433439173530821780472
