| L(s) = 1 | − 2-s + 3-s + 4-s + 2.27·5-s − 6-s − 8-s + 9-s − 2.27·10-s − 11-s + 12-s + 1.80·13-s + 2.27·15-s + 16-s + 5.10·17-s − 18-s − 6.90·19-s + 2.27·20-s + 22-s + 8.59·23-s − 24-s + 0.171·25-s − 1.80·26-s + 27-s − 2.82·29-s − 2.27·30-s + 3.14·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.01·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s − 0.719·10-s − 0.301·11-s + 0.288·12-s + 0.499·13-s + 0.587·15-s + 0.250·16-s + 1.23·17-s − 0.235·18-s − 1.58·19-s + 0.508·20-s + 0.213·22-s + 1.79·23-s − 0.204·24-s + 0.0343·25-s − 0.353·26-s + 0.192·27-s − 0.525·29-s − 0.415·30-s + 0.563·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.166756584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.166756584\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 2.27T + 5T^{2} \) |
| 13 | \( 1 - 1.80T + 13T^{2} \) |
| 17 | \( 1 - 5.10T + 17T^{2} \) |
| 19 | \( 1 + 6.90T + 19T^{2} \) |
| 23 | \( 1 - 8.59T + 23T^{2} \) |
| 29 | \( 1 + 2.82T + 29T^{2} \) |
| 31 | \( 1 - 3.14T + 31T^{2} \) |
| 37 | \( 1 - 4.44T + 37T^{2} \) |
| 41 | \( 1 + 5.10T + 41T^{2} \) |
| 43 | \( 1 - 6.93T + 43T^{2} \) |
| 47 | \( 1 + 0.0759T + 47T^{2} \) |
| 53 | \( 1 + 6.87T + 53T^{2} \) |
| 59 | \( 1 - 5.17T + 59T^{2} \) |
| 61 | \( 1 - 8.63T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 + 9.26T + 71T^{2} \) |
| 73 | \( 1 + 3.04T + 73T^{2} \) |
| 79 | \( 1 - 9.26T + 79T^{2} \) |
| 83 | \( 1 + 1.96T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 + 9.01T + 97T^{2} \) |
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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
−8.663284783434780636673066893907, −8.118226623492058775226323980550, −7.26901795195525980673382085106, −6.48276837195326434996856459618, −5.79850295649733796048217731174, −4.92871202731708751630146106394, −3.73620559926979813057613887710, −2.78472330157732655701162285431, −2.00581310805131738547428938755, −1.00170015497670842844860900267,
1.00170015497670842844860900267, 2.00581310805131738547428938755, 2.78472330157732655701162285431, 3.73620559926979813057613887710, 4.92871202731708751630146106394, 5.79850295649733796048217731174, 6.48276837195326434996856459618, 7.26901795195525980673382085106, 8.118226623492058775226323980550, 8.663284783434780636673066893907
