| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 11-s + 12-s − 1.41·13-s + 16-s − 4·17-s − 18-s − 5.41·19-s − 22-s + 7.65·23-s − 24-s − 5·25-s + 1.41·26-s + 27-s − 5.65·29-s + 3.07·31-s − 32-s + 33-s + 4·34-s + 36-s + 3.65·37-s + 5.41·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.301·11-s + 0.288·12-s − 0.392·13-s + 0.250·16-s − 0.970·17-s − 0.235·18-s − 1.24·19-s − 0.213·22-s + 1.59·23-s − 0.204·24-s − 25-s + 0.277·26-s + 0.192·27-s − 1.05·29-s + 0.551·31-s − 0.176·32-s + 0.174·33-s + 0.685·34-s + 0.166·36-s + 0.601·37-s + 0.878·38-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 5.41T + 19T^{2} \) |
| 23 | \( 1 - 7.65T + 23T^{2} \) |
| 29 | \( 1 + 5.65T + 29T^{2} \) |
| 31 | \( 1 - 3.07T + 31T^{2} \) |
| 37 | \( 1 - 3.65T + 37T^{2} \) |
| 41 | \( 1 + 9.65T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 - 1.41T + 47T^{2} \) |
| 53 | \( 1 + 3.65T + 53T^{2} \) |
| 59 | \( 1 + 6.82T + 59T^{2} \) |
| 61 | \( 1 - 1.41T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 1.65T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 + 2.58T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−8.370126173621390129695589999820, −7.70647970725627220271923738030, −6.82555087595750569363787587804, −6.39464694160007385587922893641, −5.17999560291857478239417795123, −4.30919506497167881380120618170, −3.35243553816755163813653302649, −2.38330601354905729824208190344, −1.57238200269212342711657703280, 0,
1.57238200269212342711657703280, 2.38330601354905729824208190344, 3.35243553816755163813653302649, 4.30919506497167881380120618170, 5.17999560291857478239417795123, 6.39464694160007385587922893641, 6.82555087595750569363787587804, 7.70647970725627220271923738030, 8.370126173621390129695589999820
