| L(s) = 1 | + 5.07·2-s − 5.26·3-s + 17.7·4-s − 3.42·5-s − 26.6·6-s + 25.1·7-s + 49.2·8-s + 0.724·9-s − 17.3·10-s + 69.6·11-s − 93.2·12-s + 26.4·13-s + 127.·14-s + 18.0·15-s + 107.·16-s + 87.9·17-s + 3.67·18-s + 15.6·19-s − 60.5·20-s − 132.·21-s + 353.·22-s − 0.430·23-s − 259.·24-s − 113.·25-s + 134.·26-s + 138.·27-s + 445.·28-s + ⋯ |
| L(s) = 1 | + 1.79·2-s − 1.01·3-s + 2.21·4-s − 0.305·5-s − 1.81·6-s + 1.35·7-s + 2.17·8-s + 0.0268·9-s − 0.548·10-s + 1.91·11-s − 2.24·12-s + 0.565·13-s + 2.43·14-s + 0.309·15-s + 1.68·16-s + 1.25·17-s + 0.0481·18-s + 0.188·19-s − 0.677·20-s − 1.37·21-s + 3.42·22-s − 0.00390·23-s − 2.20·24-s − 0.906·25-s + 1.01·26-s + 0.986·27-s + 3.00·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.734721003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.734721003\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 - 5.07T + 8T^{2} \) |
| 3 | \( 1 + 5.26T + 27T^{2} \) |
| 5 | \( 1 + 3.42T + 125T^{2} \) |
| 7 | \( 1 - 25.1T + 343T^{2} \) |
| 11 | \( 1 - 69.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 26.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 87.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 15.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 0.430T + 1.21e4T^{2} \) |
| 29 | \( 1 + 68.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 228.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 276.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 379.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 59.2T + 1.03e5T^{2} \) |
| 53 | \( 1 + 263.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 697.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 782.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 546.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 400.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 79.9T + 3.89e5T^{2} \) |
| 79 | \( 1 + 155.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 92.2T + 5.71e5T^{2} \) |
| 89 | \( 1 - 924.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 190.T + 9.12e5T^{2} \) |
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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
−8.728083149831674643711416158476, −7.76125972284947482980081477558, −6.91096091846764044644145921076, −6.13856440061230917730274408485, −5.53197035914355381588943320220, −4.91145345161428599010204759470, −3.98505860915300972172350209535, −3.50177680410917635673461794499, −1.91249942031156368709890766233, −1.07408997576497062879353286352,
1.07408997576497062879353286352, 1.91249942031156368709890766233, 3.50177680410917635673461794499, 3.98505860915300972172350209535, 4.91145345161428599010204759470, 5.53197035914355381588943320220, 6.13856440061230917730274408485, 6.91096091846764044644145921076, 7.76125972284947482980081477558, 8.728083149831674643711416158476
