| L(s) = 1 | + 2-s + 3-s + 4-s − 3.44·5-s + 6-s − 1.17·7-s + 8-s + 9-s − 3.44·10-s − 0.720·11-s + 12-s + 2.74·13-s − 1.17·14-s − 3.44·15-s + 16-s − 4.40·17-s + 18-s + 3.34·19-s − 3.44·20-s − 1.17·21-s − 0.720·22-s + 0.627·23-s + 24-s + 6.86·25-s + 2.74·26-s + 27-s − 1.17·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.54·5-s + 0.408·6-s − 0.443·7-s + 0.353·8-s + 0.333·9-s − 1.08·10-s − 0.217·11-s + 0.288·12-s + 0.761·13-s − 0.313·14-s − 0.889·15-s + 0.250·16-s − 1.06·17-s + 0.235·18-s + 0.766·19-s − 0.770·20-s − 0.255·21-s − 0.153·22-s + 0.130·23-s + 0.204·24-s + 1.37·25-s + 0.538·26-s + 0.192·27-s − 0.221·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5766 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5766 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 5 | \( 1 + 3.44T + 5T^{2} \) |
| 7 | \( 1 + 1.17T + 7T^{2} \) |
| 11 | \( 1 + 0.720T + 11T^{2} \) |
| 13 | \( 1 - 2.74T + 13T^{2} \) |
| 17 | \( 1 + 4.40T + 17T^{2} \) |
| 19 | \( 1 - 3.34T + 19T^{2} \) |
| 23 | \( 1 - 0.627T + 23T^{2} \) |
| 29 | \( 1 - 2.84T + 29T^{2} \) |
| 37 | \( 1 + 11.1T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 5.14T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 + 3.58T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 + 5.04T + 71T^{2} \) |
| 73 | \( 1 - 0.479T + 73T^{2} \) |
| 79 | \( 1 - 1.45T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 + 2.44T + 97T^{2} \) |
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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
−7.67404475522218620841817392205, −7.05570986019467886435614746024, −6.51962342516656147745394260669, −5.46950235670480571627276720213, −4.60498560010003164625746757536, −3.98332148907310074258940580935, −3.35541928939274353140206908350, −2.75882743545526640879197384021, −1.45468044805210522493261786601, 0,
1.45468044805210522493261786601, 2.75882743545526640879197384021, 3.35541928939274353140206908350, 3.98332148907310074258940580935, 4.60498560010003164625746757536, 5.46950235670480571627276720213, 6.51962342516656147745394260669, 7.05570986019467886435614746024, 7.67404475522218620841817392205
