| L(s) = 1 | + 2-s − 0.953·3-s + 4-s − 3.75·5-s − 0.953·6-s − 3.81·7-s + 8-s − 2.08·9-s − 3.75·10-s + 5.51·11-s − 0.953·12-s − 3.81·14-s + 3.58·15-s + 16-s + 4.01·17-s − 2.08·18-s − 19-s − 3.75·20-s + 3.63·21-s + 5.51·22-s − 8.48·23-s − 0.953·24-s + 9.09·25-s + 4.85·27-s − 3.81·28-s + 3.65·29-s + 3.58·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.550·3-s + 0.5·4-s − 1.67·5-s − 0.389·6-s − 1.44·7-s + 0.353·8-s − 0.696·9-s − 1.18·10-s + 1.66·11-s − 0.275·12-s − 1.01·14-s + 0.924·15-s + 0.250·16-s + 0.973·17-s − 0.492·18-s − 0.229·19-s − 0.839·20-s + 0.793·21-s + 1.17·22-s − 1.76·23-s − 0.194·24-s + 1.81·25-s + 0.934·27-s − 0.720·28-s + 0.677·29-s + 0.653·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 0.953T + 3T^{2} \) |
| 5 | \( 1 + 3.75T + 5T^{2} \) |
| 7 | \( 1 + 3.81T + 7T^{2} \) |
| 11 | \( 1 - 5.51T + 11T^{2} \) |
| 17 | \( 1 - 4.01T + 17T^{2} \) |
| 23 | \( 1 + 8.48T + 23T^{2} \) |
| 29 | \( 1 - 3.65T + 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 - 0.902T + 37T^{2} \) |
| 41 | \( 1 + 4.14T + 41T^{2} \) |
| 43 | \( 1 + 5.76T + 43T^{2} \) |
| 47 | \( 1 - 6.37T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 0.808T + 59T^{2} \) |
| 61 | \( 1 + 1.66T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 + 5.59T + 71T^{2} \) |
| 73 | \( 1 + 2.82T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−7.51040396799353577693409450906, −6.68547393678156147933243822543, −6.33421500209135320118884199923, −5.68615596740490397578384611235, −4.53332380429539862528957437798, −3.99017162702242996053778078958, −3.41700257205770116604033567111, −2.75826272711718265214517205024, −1.04644444110285706947054059723, 0,
1.04644444110285706947054059723, 2.75826272711718265214517205024, 3.41700257205770116604033567111, 3.99017162702242996053778078958, 4.53332380429539862528957437798, 5.68615596740490397578384611235, 6.33421500209135320118884199923, 6.68547393678156147933243822543, 7.51040396799353577693409450906
