| L(s) = 1 | − 3·3-s − 19.4·5-s + 6.74·7-s + 9·9-s − 11·11-s + 60.9·13-s + 58.4·15-s − 99.1·17-s − 24.7·19-s − 20.2·21-s + 112·23-s + 254.·25-s − 27·27-s + 21.1·29-s − 318.·31-s + 33·33-s − 131.·35-s + 150.·37-s − 182.·39-s − 252.·41-s − 214.·43-s − 175.·45-s + 105.·47-s − 297.·49-s + 297.·51-s − 325.·53-s + 214.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.74·5-s + 0.364·7-s + 0.333·9-s − 0.301·11-s + 1.30·13-s + 1.00·15-s − 1.41·17-s − 0.298·19-s − 0.210·21-s + 1.01·23-s + 2.03·25-s − 0.192·27-s + 0.135·29-s − 1.84·31-s + 0.174·33-s − 0.634·35-s + 0.668·37-s − 0.751·39-s − 0.962·41-s − 0.759·43-s − 0.581·45-s + 0.328·47-s − 0.867·49-s + 0.816·51-s − 0.843·53-s + 0.525·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6662185939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6662185939\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 11 | \( 1 + 11T \) |
| good | 5 | \( 1 + 19.4T + 125T^{2} \) |
| 7 | \( 1 - 6.74T + 343T^{2} \) |
| 13 | \( 1 - 60.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 99.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 24.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 112T + 1.21e4T^{2} \) |
| 29 | \( 1 - 21.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 318.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 150.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 252.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 214.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 105.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 325.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 196T + 2.05e5T^{2} \) |
| 61 | \( 1 - 402.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 27.4T + 3.00e5T^{2} \) |
| 71 | \( 1 + 300.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 427.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 97.5T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.10e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 463.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.79e3T + 9.12e5T^{2} \) |
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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
−8.518844064734792291814273750273, −8.079170676970730843229387032713, −7.09770802729487309605554086625, −6.61990733007658328935094938459, −5.45385926001313293068438342322, −4.58361105548085528070237480827, −3.96717075587333736444893185392, −3.11460794116014036098633868459, −1.59592044525042769911340118127, −0.39155561342061303224272842288,
0.39155561342061303224272842288, 1.59592044525042769911340118127, 3.11460794116014036098633868459, 3.96717075587333736444893185392, 4.58361105548085528070237480827, 5.45385926001313293068438342322, 6.61990733007658328935094938459, 7.09770802729487309605554086625, 8.079170676970730843229387032713, 8.518844064734792291814273750273
