| L(s) = 1 | + 2-s − 3-s + 4-s − 3·5-s − 6-s − 7-s + 8-s + 9-s − 3·10-s − 12-s + 5·13-s − 14-s + 3·15-s + 16-s − 17-s + 18-s + 4·19-s − 3·20-s + 21-s + 23-s − 24-s + 4·25-s + 5·26-s − 27-s − 28-s + 3·29-s + 3·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.948·10-s − 0.288·12-s + 1.38·13-s − 0.267·14-s + 0.774·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.917·19-s − 0.670·20-s + 0.218·21-s + 0.208·23-s − 0.204·24-s + 4/5·25-s + 0.980·26-s − 0.192·27-s − 0.188·28-s + 0.557·29-s + 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.719313101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.719313101\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−13.47622570084383, −13.10394573358984, −12.51931099846226, −12.18121684666117, −11.66283937787180, −11.26157062808522, −10.83352123724963, −10.58895329475154, −9.518631595930430, −9.409035256724563, −8.439819276231656, −8.092292333110322, −7.580018650892045, −7.015705066097974, −6.520693211142543, −6.057891631533723, −5.501513342342407, −4.864796449673989, −4.354695626078075, −3.816611321354422, −3.403392887253485, −2.889685923556517, −1.945745226127139, −1.059862692647025, −0.5505002948283533,
0.5505002948283533, 1.059862692647025, 1.945745226127139, 2.889685923556517, 3.403392887253485, 3.816611321354422, 4.354695626078075, 4.864796449673989, 5.501513342342407, 6.057891631533723, 6.520693211142543, 7.015705066097974, 7.580018650892045, 8.092292333110322, 8.439819276231656, 9.409035256724563, 9.518631595930430, 10.58895329475154, 10.83352123724963, 11.26157062808522, 11.66283937787180, 12.18121684666117, 12.51931099846226, 13.10394573358984, 13.47622570084383
