| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 3.52·7-s − 8-s + 9-s + 5.25·11-s − 12-s − 0.619·13-s − 3.52·14-s + 16-s − 3.44·17-s − 18-s + 2.27·19-s − 3.52·21-s − 5.25·22-s + 8.95·23-s + 24-s + 0.619·26-s − 27-s + 3.52·28-s − 2.68·29-s + 9.76·31-s − 32-s − 5.25·33-s + 3.44·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s + 1.33·7-s − 0.353·8-s + 0.333·9-s + 1.58·11-s − 0.288·12-s − 0.171·13-s − 0.941·14-s + 0.250·16-s − 0.835·17-s − 0.235·18-s + 0.521·19-s − 0.768·21-s − 1.12·22-s + 1.86·23-s + 0.204·24-s + 0.121·26-s − 0.192·27-s + 0.665·28-s − 0.497·29-s + 1.75·31-s − 0.176·32-s − 0.914·33-s + 0.590·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.586766939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.586766939\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 3.52T + 7T^{2} \) |
| 11 | \( 1 - 5.25T + 11T^{2} \) |
| 13 | \( 1 + 0.619T + 13T^{2} \) |
| 17 | \( 1 + 3.44T + 17T^{2} \) |
| 19 | \( 1 - 2.27T + 19T^{2} \) |
| 23 | \( 1 - 8.95T + 23T^{2} \) |
| 29 | \( 1 + 2.68T + 29T^{2} \) |
| 31 | \( 1 - 9.76T + 31T^{2} \) |
| 37 | \( 1 - 1.00T + 37T^{2} \) |
| 41 | \( 1 - 6.29T + 41T^{2} \) |
| 43 | \( 1 - 1.51T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + 0.553T + 53T^{2} \) |
| 59 | \( 1 - 0.278T + 59T^{2} \) |
| 61 | \( 1 - 3.68T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 - 4.74T + 71T^{2} \) |
| 73 | \( 1 + 9.83T + 73T^{2} \) |
| 79 | \( 1 - 6.52T + 79T^{2} \) |
| 83 | \( 1 + 2.65T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + 4.37T + 97T^{2} \) |
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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
−8.502914681103929140112778135209, −7.88767740145923799787626344778, −6.88877324470737021411325217404, −6.62595342699649202641893658830, −5.52059152220800558485862716221, −4.75670985409453096161123168318, −4.06229043309977308203186308105, −2.77160724666157675633142735290, −1.58064045692853118509980946767, −0.943408881415150737903748193077,
0.943408881415150737903748193077, 1.58064045692853118509980946767, 2.77160724666157675633142735290, 4.06229043309977308203186308105, 4.75670985409453096161123168318, 5.52059152220800558485862716221, 6.62595342699649202641893658830, 6.88877324470737021411325217404, 7.88767740145923799787626344778, 8.502914681103929140112778135209
