| L(s) = 1 | − 2.32·2-s + 9.13·3-s − 2.61·4-s + 12.9·5-s − 21.1·6-s − 21.6·7-s + 24.6·8-s + 56.4·9-s − 30.0·10-s − 29.2·11-s − 23.9·12-s + 2.67·13-s + 50.1·14-s + 118.·15-s − 36.2·16-s − 137.·17-s − 131.·18-s + 51.9·19-s − 33.9·20-s − 197.·21-s + 67.8·22-s + 92.6·23-s + 225.·24-s + 43.0·25-s − 6.20·26-s + 269.·27-s + 56.5·28-s + ⋯ |
| L(s) = 1 | − 0.820·2-s + 1.75·3-s − 0.327·4-s + 1.15·5-s − 1.44·6-s − 1.16·7-s + 1.08·8-s + 2.09·9-s − 0.950·10-s − 0.801·11-s − 0.575·12-s + 0.0570·13-s + 0.957·14-s + 2.03·15-s − 0.565·16-s − 1.95·17-s − 1.71·18-s + 0.627·19-s − 0.379·20-s − 2.05·21-s + 0.657·22-s + 0.840·23-s + 1.91·24-s + 0.344·25-s − 0.0467·26-s + 1.91·27-s + 0.381·28-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 2.32T + 8T^{2} \) |
| 3 | \( 1 - 9.13T + 27T^{2} \) |
| 5 | \( 1 - 12.9T + 125T^{2} \) |
| 7 | \( 1 + 21.6T + 343T^{2} \) |
| 11 | \( 1 + 29.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 2.67T + 2.19e3T^{2} \) |
| 17 | \( 1 + 137.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 51.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 92.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 148.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 130.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 211.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 474.T + 6.89e4T^{2} \) |
| 47 | \( 1 - 174.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 543.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 85.7T + 2.05e5T^{2} \) |
| 61 | \( 1 + 238.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 316.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 51.1T + 3.57e5T^{2} \) |
| 73 | \( 1 - 219.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.17e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 921.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 910.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 884.T + 9.12e5T^{2} \) |
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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
−8.727378143250581854162590848857, −8.007629971599270704104749426037, −7.14456905191736014342975530091, −6.43278629542226099927337497379, −5.13627787627830132268956236966, −4.15720666455233696320225560799, −3.08243661656579386576290621145, −2.38001727738204969431985730139, −1.50341701671064959257445275026, 0,
1.50341701671064959257445275026, 2.38001727738204969431985730139, 3.08243661656579386576290621145, 4.15720666455233696320225560799, 5.13627787627830132268956236966, 6.43278629542226099927337497379, 7.14456905191736014342975530091, 8.007629971599270704104749426037, 8.727378143250581854162590848857
