| L(s) = 1 | − 2-s − 3-s + 4-s − 3.70·5-s + 6-s − 8-s + 9-s + 3.70·10-s − 4·11-s − 12-s − 5.70·13-s + 3.70·15-s + 16-s − 4·17-s − 18-s + 5.40·19-s − 3.70·20-s + 4·22-s + 23-s + 24-s + 8.70·25-s + 5.70·26-s − 27-s + 0.298·29-s − 3.70·30-s + 2·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.65·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s + 1.17·10-s − 1.20·11-s − 0.288·12-s − 1.58·13-s + 0.955·15-s + 0.250·16-s − 0.970·17-s − 0.235·18-s + 1.23·19-s − 0.827·20-s + 0.852·22-s + 0.208·23-s + 0.204·24-s + 1.74·25-s + 1.11·26-s − 0.192·27-s + 0.0554·29-s − 0.675·30-s + 0.359·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 3.70T + 5T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 5.70T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 5.40T + 19T^{2} \) |
| 29 | \( 1 - 0.298T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 4.29T + 37T^{2} \) |
| 41 | \( 1 - 0.298T + 41T^{2} \) |
| 43 | \( 1 + 1.70T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + 9.40T + 53T^{2} \) |
| 59 | \( 1 - 7.40T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 + 7.40T + 71T^{2} \) |
| 73 | \( 1 + 1.40T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 + 6.29T + 97T^{2} \) |
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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
−7.49191662657071870712973170466, −7.33405043483349563643895621019, −6.48272235791769674930536957301, −5.33810153645552896380643486623, −4.84692657850520050187422770260, −4.06563861416579367207014785127, −3.02708975877617637456645976540, −2.34131916182092945441788993781, −0.77228464635059305371408620502, 0,
0.77228464635059305371408620502, 2.34131916182092945441788993781, 3.02708975877617637456645976540, 4.06563861416579367207014785127, 4.84692657850520050187422770260, 5.33810153645552896380643486623, 6.48272235791769674930536957301, 7.33405043483349563643895621019, 7.49191662657071870712973170466
