| L(s) = 1 | − 1.39·2-s + 8.98·3-s − 6.06·4-s − 6.95·5-s − 12.4·6-s − 9.63·7-s + 19.5·8-s + 53.6·9-s + 9.67·10-s − 11·11-s − 54.4·12-s + 13.3·14-s − 62.4·15-s + 21.3·16-s + 29.0·17-s − 74.5·18-s + 0.463·19-s + 42.2·20-s − 86.5·21-s + 15.2·22-s − 162.·23-s + 175.·24-s − 76.6·25-s + 239.·27-s + 58.4·28-s + 21.5·29-s + 86.8·30-s + ⋯ |
| L(s) = 1 | − 0.491·2-s + 1.72·3-s − 0.758·4-s − 0.622·5-s − 0.849·6-s − 0.520·7-s + 0.864·8-s + 1.98·9-s + 0.305·10-s − 0.301·11-s − 1.31·12-s + 0.255·14-s − 1.07·15-s + 0.333·16-s + 0.415·17-s − 0.976·18-s + 0.00559·19-s + 0.471·20-s − 0.899·21-s + 0.148·22-s − 1.47·23-s + 1.49·24-s − 0.612·25-s + 1.70·27-s + 0.394·28-s + 0.137·29-s + 0.528·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.953559734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.953559734\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.39T + 8T^{2} \) |
| 3 | \( 1 - 8.98T + 27T^{2} \) |
| 5 | \( 1 + 6.95T + 125T^{2} \) |
| 7 | \( 1 + 9.63T + 343T^{2} \) |
| 17 | \( 1 - 29.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 0.463T + 6.85e3T^{2} \) |
| 23 | \( 1 + 162.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 21.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 150.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 21.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 12.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 199.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 395.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 590.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 66.0T + 2.05e5T^{2} \) |
| 61 | \( 1 + 505.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 198.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 863.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 981.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 131.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 713.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.35e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 56.4T + 9.12e5T^{2} \) |
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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
−8.782285946280186612497650945498, −8.054419925858812639164888089448, −7.86153373364507441397020281975, −6.92536631011938375091228118459, −5.61991335043699411486120777294, −4.28732303583426078959379988033, −3.89333953398977890065337332249, −2.97146401906975748060671448792, −1.95673796018640755848089343447, −0.64602209567014149453714795269,
0.64602209567014149453714795269, 1.95673796018640755848089343447, 2.97146401906975748060671448792, 3.89333953398977890065337332249, 4.28732303583426078959379988033, 5.61991335043699411486120777294, 6.92536631011938375091228118459, 7.86153373364507441397020281975, 8.054419925858812639164888089448, 8.782285946280186612497650945498
