| L(s) = 1 | − 3-s + 0.614·5-s + 2.26·7-s + 9-s + 6.42·11-s − 6.12·13-s − 0.614·15-s + 5.70·17-s − 5.62·19-s − 2.26·21-s − 6.95·23-s − 4.62·25-s − 27-s + 8.67·29-s − 9.66·31-s − 6.42·33-s + 1.38·35-s − 5.18·37-s + 6.12·39-s − 3.34·41-s − 1.33·43-s + 0.614·45-s + 3.17·47-s − 1.88·49-s − 5.70·51-s − 5.99·53-s + 3.94·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.274·5-s + 0.854·7-s + 0.333·9-s + 1.93·11-s − 1.69·13-s − 0.158·15-s + 1.38·17-s − 1.29·19-s − 0.493·21-s − 1.45·23-s − 0.924·25-s − 0.192·27-s + 1.61·29-s − 1.73·31-s − 1.11·33-s + 0.234·35-s − 0.852·37-s + 0.980·39-s − 0.521·41-s − 0.204·43-s + 0.0916·45-s + 0.462·47-s − 0.269·49-s − 0.799·51-s − 0.823·53-s + 0.532·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 251 | \( 1 + T \) |
| good | 5 | \( 1 - 0.614T + 5T^{2} \) |
| 7 | \( 1 - 2.26T + 7T^{2} \) |
| 11 | \( 1 - 6.42T + 11T^{2} \) |
| 13 | \( 1 + 6.12T + 13T^{2} \) |
| 17 | \( 1 - 5.70T + 17T^{2} \) |
| 19 | \( 1 + 5.62T + 19T^{2} \) |
| 23 | \( 1 + 6.95T + 23T^{2} \) |
| 29 | \( 1 - 8.67T + 29T^{2} \) |
| 31 | \( 1 + 9.66T + 31T^{2} \) |
| 37 | \( 1 + 5.18T + 37T^{2} \) |
| 41 | \( 1 + 3.34T + 41T^{2} \) |
| 43 | \( 1 + 1.33T + 43T^{2} \) |
| 47 | \( 1 - 3.17T + 47T^{2} \) |
| 53 | \( 1 + 5.99T + 53T^{2} \) |
| 59 | \( 1 + 5.78T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 - 3.64T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 - 5.59T + 79T^{2} \) |
| 83 | \( 1 + 3.01T + 83T^{2} \) |
| 89 | \( 1 + 0.395T + 89T^{2} \) |
| 97 | \( 1 + 1.97T + 97T^{2} \) |
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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
−7.63455225126820965081337515307, −7.01519509749670066685595024081, −6.20163069946584025805236880266, −5.69203510212636189780973478470, −4.71951011593493421268924144708, −4.28706354003688045965663275009, −3.33861394545048355029544755647, −1.96380776463375518423642676145, −1.50343807791818533901210754063, 0,
1.50343807791818533901210754063, 1.96380776463375518423642676145, 3.33861394545048355029544755647, 4.28706354003688045965663275009, 4.71951011593493421268924144708, 5.69203510212636189780973478470, 6.20163069946584025805236880266, 7.01519509749670066685595024081, 7.63455225126820965081337515307
