| L(s) = 1 | + 2-s − 3-s + 4-s − 0.454·5-s − 6-s + 8-s + 9-s − 0.454·10-s + 11-s − 12-s − 0.909·13-s + 0.454·15-s + 16-s − 17-s + 18-s − 3.88·19-s − 0.454·20-s + 22-s + 8.33·23-s − 24-s − 4.79·25-s − 0.909·26-s − 27-s − 2.79·29-s + 0.454·30-s + 9.58·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.203·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.143·10-s + 0.301·11-s − 0.288·12-s − 0.252·13-s + 0.117·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s − 0.890·19-s − 0.101·20-s + 0.213·22-s + 1.73·23-s − 0.204·24-s − 0.958·25-s − 0.178·26-s − 0.192·27-s − 0.518·29-s + 0.0830·30-s + 1.72·31-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
| \(L(1)\) |
\(\approx\) |
\(2.316636865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.316636865\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.454T + 5T^{2} \) |
| 13 | \( 1 + 0.909T + 13T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 19 | \( 1 + 3.88T + 19T^{2} \) |
| 23 | \( 1 - 8.33T + 23T^{2} \) |
| 29 | \( 1 + 2.79T + 29T^{2} \) |
| 31 | \( 1 - 9.58T + 31T^{2} \) |
| 37 | \( 1 - 7.88T + 37T^{2} \) |
| 41 | \( 1 - 1.79T + 41T^{2} \) |
| 43 | \( 1 - 7.88T + 43T^{2} \) |
| 47 | \( 1 + 0.338T + 47T^{2} \) |
| 53 | \( 1 - 0.909T + 53T^{2} \) |
| 59 | \( 1 + 6.97T + 59T^{2} \) |
| 61 | \( 1 - 14.0T + 61T^{2} \) |
| 67 | \( 1 + 3.79T + 67T^{2} \) |
| 71 | \( 1 + 4.79T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 - 5.36T + 79T^{2} \) |
| 83 | \( 1 + 1.97T + 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−8.612199313393987921059387863932, −7.69280876144120612168713386256, −7.01634175439641223572151640372, −6.26441365391880491461536260210, −5.67755008610389425682331262641, −4.63549863711891001640384890407, −4.26449030872215427077904287456, −3.13906998293170211770781339160, −2.18105242605040375493198853465, −0.854915983329421438109737707473,
0.854915983329421438109737707473, 2.18105242605040375493198853465, 3.13906998293170211770781339160, 4.26449030872215427077904287456, 4.63549863711891001640384890407, 5.67755008610389425682331262641, 6.26441365391880491461536260210, 7.01634175439641223572151640372, 7.69280876144120612168713386256, 8.612199313393987921059387863932
