| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s − 0.404·11-s + 12-s − 4.12·13-s − 15-s + 16-s − 5.31·17-s − 18-s + 2.71·19-s − 20-s + 0.404·22-s − 23-s − 24-s − 4·25-s + 4.12·26-s + 27-s + 9.12·29-s + 30-s + 9.83·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.121·11-s + 0.288·12-s − 1.14·13-s − 0.258·15-s + 0.250·16-s − 1.28·17-s − 0.235·18-s + 0.623·19-s − 0.223·20-s + 0.0862·22-s − 0.208·23-s − 0.204·24-s − 0.800·25-s + 0.808·26-s + 0.192·27-s + 1.69·29-s + 0.182·30-s + 1.76·31-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 | \( 1 + T + 5T^{2} \) |
| 11 | \( 1 + 0.404T + 11T^{2} \) |
| 13 | \( 1 + 4.12T + 13T^{2} \) |
| 17 | \( 1 + 5.31T + 17T^{2} \) |
| 19 | \( 1 - 2.71T + 19T^{2} \) |
| 29 | \( 1 - 9.12T + 29T^{2} \) |
| 31 | \( 1 - 9.83T + 31T^{2} \) |
| 37 | \( 1 + 7.90T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 - 0.808T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 + 0.191T + 53T^{2} \) |
| 59 | \( 1 - 8.31T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 2.83T + 67T^{2} \) |
| 71 | \( 1 + 6.92T + 71T^{2} \) |
| 73 | \( 1 + 4.59T + 73T^{2} \) |
| 79 | \( 1 + 9.92T + 79T^{2} \) |
| 83 | \( 1 - 4.40T + 83T^{2} \) |
| 89 | \( 1 + 7.90T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−7.77406838566217804838595446354, −7.08497648819684300969455536288, −6.54299172064182605358818132443, −5.55425206344771554238505584651, −4.57096024048729199789663386827, −4.04187144817444009989973873588, −2.74506833880287681186159289538, −2.50274142739295888620201324250, −1.21639286949392565591633363546, 0,
1.21639286949392565591633363546, 2.50274142739295888620201324250, 2.74506833880287681186159289538, 4.04187144817444009989973873588, 4.57096024048729199789663386827, 5.55425206344771554238505584651, 6.54299172064182605358818132443, 7.08497648819684300969455536288, 7.77406838566217804838595446354
