| L(s) = 1 | + 2-s − 3.26·3-s + 4-s − 5-s − 3.26·6-s + 2.73·7-s + 8-s + 7.64·9-s − 10-s + 11-s − 3.26·12-s + 6.11·13-s + 2.73·14-s + 3.26·15-s + 16-s + 7.48·17-s + 7.64·18-s − 7.78·19-s − 20-s − 8.91·21-s + 22-s + 8.88·23-s − 3.26·24-s + 25-s + 6.11·26-s − 15.1·27-s + 2.73·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.88·3-s + 0.5·4-s − 0.447·5-s − 1.33·6-s + 1.03·7-s + 0.353·8-s + 2.54·9-s − 0.316·10-s + 0.301·11-s − 0.941·12-s + 1.69·13-s + 0.730·14-s + 0.842·15-s + 0.250·16-s + 1.81·17-s + 1.80·18-s − 1.78·19-s − 0.223·20-s − 1.94·21-s + 0.213·22-s + 1.85·23-s − 0.665·24-s + 0.200·25-s + 1.19·26-s − 2.91·27-s + 0.516·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.236440121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.236440121\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 + 3.26T + 3T^{2} \) |
| 7 | \( 1 - 2.73T + 7T^{2} \) |
| 13 | \( 1 - 6.11T + 13T^{2} \) |
| 17 | \( 1 - 7.48T + 17T^{2} \) |
| 19 | \( 1 + 7.78T + 19T^{2} \) |
| 23 | \( 1 - 8.88T + 23T^{2} \) |
| 29 | \( 1 - 6.36T + 29T^{2} \) |
| 31 | \( 1 - 6.01T + 31T^{2} \) |
| 37 | \( 1 + 0.607T + 37T^{2} \) |
| 41 | \( 1 - 2.66T + 41T^{2} \) |
| 47 | \( 1 - 0.852T + 47T^{2} \) |
| 53 | \( 1 + 9.54T + 53T^{2} \) |
| 59 | \( 1 + 1.12T + 59T^{2} \) |
| 61 | \( 1 - 9.95T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 + 4.13T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 + 2.52T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 - 1.91T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−8.121151455116124530730966933299, −7.31620368064106851508019399461, −6.49560591002883483264397188158, −6.08827733780176826602407272534, −5.34570957508983747143985002915, −4.62186097166159386515040561096, −4.20847985328712505572730460897, −3.16006526695854062359370712534, −1.45469265277662554040794008604, −0.969721852380763681807924990343,
0.969721852380763681807924990343, 1.45469265277662554040794008604, 3.16006526695854062359370712534, 4.20847985328712505572730460897, 4.62186097166159386515040561096, 5.34570957508983747143985002915, 6.08827733780176826602407272534, 6.49560591002883483264397188158, 7.31620368064106851508019399461, 8.121151455116124530730966933299
