| L(s) = 1 | − 3-s + 1.67·7-s + 9-s + 5.95·11-s − 5.95·13-s − 2.05·17-s − 7.00·19-s − 1.67·21-s − 2.58·23-s − 27-s + 0.763·29-s − 3.29·31-s − 5.95·33-s − 2.44·37-s + 5.95·39-s − 0.441·41-s + 1.35·43-s − 7.57·47-s − 4.18·49-s + 2.05·51-s + 9.08·53-s + 7.00·57-s − 3.55·59-s + 10.4·61-s + 1.67·63-s + 5.90·67-s + 2.58·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.634·7-s + 0.333·9-s + 1.79·11-s − 1.65·13-s − 0.499·17-s − 1.60·19-s − 0.366·21-s − 0.538·23-s − 0.192·27-s + 0.141·29-s − 0.591·31-s − 1.03·33-s − 0.401·37-s + 0.952·39-s − 0.0689·41-s + 0.206·43-s − 1.10·47-s − 0.598·49-s + 0.288·51-s + 1.24·53-s + 0.928·57-s − 0.463·59-s + 1.34·61-s + 0.211·63-s + 0.720·67-s + 0.310·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 1.67T + 7T^{2} \) |
| 11 | \( 1 - 5.95T + 11T^{2} \) |
| 13 | \( 1 + 5.95T + 13T^{2} \) |
| 17 | \( 1 + 2.05T + 17T^{2} \) |
| 19 | \( 1 + 7.00T + 19T^{2} \) |
| 23 | \( 1 + 2.58T + 23T^{2} \) |
| 29 | \( 1 - 0.763T + 29T^{2} \) |
| 31 | \( 1 + 3.29T + 31T^{2} \) |
| 37 | \( 1 + 2.44T + 37T^{2} \) |
| 41 | \( 1 + 0.441T + 41T^{2} \) |
| 43 | \( 1 - 1.35T + 43T^{2} \) |
| 47 | \( 1 + 7.57T + 47T^{2} \) |
| 53 | \( 1 - 9.08T + 53T^{2} \) |
| 59 | \( 1 + 3.55T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 - 5.90T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 8.99T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−8.450854955180045358489100179996, −7.45716143699109983692596234692, −6.76814821532493916517288871067, −6.21328720959264233037555706538, −5.17345934392728006634333238135, −4.45612978025741787053799712115, −3.85627876209933024488957601807, −2.35611139115719336770726865722, −1.52872389903784295322367948215, 0,
1.52872389903784295322367948215, 2.35611139115719336770726865722, 3.85627876209933024488957601807, 4.45612978025741787053799712115, 5.17345934392728006634333238135, 6.21328720959264233037555706538, 6.76814821532493916517288871067, 7.45716143699109983692596234692, 8.450854955180045358489100179996
