| L(s) = 1 | − 2.24·2-s − 1.84·3-s − 2.97·4-s + 18.3·5-s + 4.13·6-s − 16.8·7-s + 24.6·8-s − 23.6·9-s − 41.0·10-s − 52.4·11-s + 5.48·12-s − 87.5·13-s + 37.7·14-s − 33.7·15-s − 31.3·16-s − 15.4·17-s + 52.8·18-s − 67.0·19-s − 54.5·20-s + 31.0·21-s + 117.·22-s + 132.·23-s − 45.3·24-s + 211.·25-s + 196.·26-s + 93.2·27-s + 50.1·28-s + ⋯ |
| L(s) = 1 | − 0.792·2-s − 0.354·3-s − 0.372·4-s + 1.63·5-s + 0.281·6-s − 0.910·7-s + 1.08·8-s − 0.874·9-s − 1.29·10-s − 1.43·11-s + 0.131·12-s − 1.86·13-s + 0.721·14-s − 0.581·15-s − 0.489·16-s − 0.219·17-s + 0.692·18-s − 0.809·19-s − 0.610·20-s + 0.322·21-s + 1.13·22-s + 1.20·23-s − 0.385·24-s + 1.68·25-s + 1.48·26-s + 0.664·27-s + 0.338·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4609280653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4609280653\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 \) |
| good | 2 | \( 1 + 2.24T + 8T^{2} \) |
| 3 | \( 1 + 1.84T + 27T^{2} \) |
| 5 | \( 1 - 18.3T + 125T^{2} \) |
| 7 | \( 1 + 16.8T + 343T^{2} \) |
| 11 | \( 1 + 52.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 87.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 15.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 67.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 132.T + 1.21e4T^{2} \) |
| 31 | \( 1 + 90.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 11.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 18.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 147.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 21.0T + 1.03e5T^{2} \) |
| 53 | \( 1 + 290.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 337.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 84.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 330.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 492.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 347.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 986.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 594.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.38e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 334.T + 9.12e5T^{2} \) |
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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
−9.603170417325091184188325572541, −9.347709484567319446855559090924, −8.311712206003296780808104628948, −7.27079667472659565948578635659, −6.33953972887126697767669007617, −5.31606330774829179418312344327, −4.89973955408423919201376423191, −2.87611726055021527085811938561, −2.13695941380893191057998291401, −0.40581220571309589628449162823,
0.40581220571309589628449162823, 2.13695941380893191057998291401, 2.87611726055021527085811938561, 4.89973955408423919201376423191, 5.31606330774829179418312344327, 6.33953972887126697767669007617, 7.27079667472659565948578635659, 8.311712206003296780808104628948, 9.347709484567319446855559090924, 9.603170417325091184188325572541
