| L(s) = 1 | − 2-s − 3-s + 4-s + 2.64·5-s + 6-s − 8-s + 9-s − 2.64·10-s + 11-s − 12-s − 4·13-s − 2.64·15-s + 16-s − 3·17-s − 18-s + 5.29·19-s + 2.64·20-s − 22-s − 2.64·23-s + 24-s + 2.00·25-s + 4·26-s − 27-s + 2·29-s + 2.64·30-s − 4·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.18·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s − 0.836·10-s + 0.301·11-s − 0.288·12-s − 1.10·13-s − 0.683·15-s + 0.250·16-s − 0.727·17-s − 0.235·18-s + 1.21·19-s + 0.591·20-s − 0.213·22-s − 0.551·23-s + 0.204·24-s + 0.400·25-s + 0.784·26-s − 0.192·27-s + 0.371·29-s + 0.483·30-s − 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.278680455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.278680455\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 2.64T + 5T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 5.29T + 19T^{2} \) |
| 23 | \( 1 + 2.64T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 1.29T + 37T^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 - 9.29T + 43T^{2} \) |
| 47 | \( 1 + 3.93T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 - 6.58T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 - 7.58T + 67T^{2} \) |
| 71 | \( 1 + 2.70T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 - 2.70T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−8.917605534230616761819348028343, −7.80265717588131564985471519427, −7.17297204238244352029051710425, −6.42693823002485659828964369960, −5.70882391913891838011310476851, −5.10206233197082223618877274641, −4.01831539051829615806540306675, −2.64347178413780953086229347103, −1.95048646589176865685364307025, −0.77817346615525128588605157229,
0.77817346615525128588605157229, 1.95048646589176865685364307025, 2.64347178413780953086229347103, 4.01831539051829615806540306675, 5.10206233197082223618877274641, 5.70882391913891838011310476851, 6.42693823002485659828964369960, 7.17297204238244352029051710425, 7.80265717588131564985471519427, 8.917605534230616761819348028343
