L(s) = 1 | − 2.65·2-s + 2.92·3-s + 5.06·4-s + 0.928·5-s − 7.76·6-s + 3.41·7-s − 8.13·8-s + 5.53·9-s − 2.46·10-s + 3.81·11-s + 14.7·12-s + 0.216·13-s − 9.08·14-s + 2.71·15-s + 11.4·16-s − 1.11·17-s − 14.7·18-s + 2.30·19-s + 4.69·20-s + 9.98·21-s − 10.1·22-s + 9.12·23-s − 23.7·24-s − 4.13·25-s − 0.575·26-s + 7.39·27-s + 17.2·28-s + ⋯ |
L(s) = 1 | − 1.87·2-s + 1.68·3-s + 2.53·4-s + 0.415·5-s − 3.16·6-s + 1.29·7-s − 2.87·8-s + 1.84·9-s − 0.780·10-s + 1.14·11-s + 4.26·12-s + 0.0601·13-s − 2.42·14-s + 0.700·15-s + 2.87·16-s − 0.269·17-s − 3.46·18-s + 0.528·19-s + 1.05·20-s + 2.17·21-s − 2.15·22-s + 1.90·23-s − 4.84·24-s − 0.827·25-s − 0.112·26-s + 1.42·27-s + 3.26·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.482880645\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.482880645\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 6037 | \( 1+O(T) \) |
good | 2 | \( 1 + 2.65T + 2T^{2} \) |
| 3 | \( 1 - 2.92T + 3T^{2} \) |
| 5 | \( 1 - 0.928T + 5T^{2} \) |
| 7 | \( 1 - 3.41T + 7T^{2} \) |
| 11 | \( 1 - 3.81T + 11T^{2} \) |
| 13 | \( 1 - 0.216T + 13T^{2} \) |
| 17 | \( 1 + 1.11T + 17T^{2} \) |
| 19 | \( 1 - 2.30T + 19T^{2} \) |
| 23 | \( 1 - 9.12T + 23T^{2} \) |
| 29 | \( 1 - 0.567T + 29T^{2} \) |
| 31 | \( 1 + 8.02T + 31T^{2} \) |
| 37 | \( 1 + 4.36T + 37T^{2} \) |
| 41 | \( 1 - 3.05T + 41T^{2} \) |
| 43 | \( 1 - 7.90T + 43T^{2} \) |
| 47 | \( 1 - 6.04T + 47T^{2} \) |
| 53 | \( 1 + 8.67T + 53T^{2} \) |
| 59 | \( 1 + 0.276T + 59T^{2} \) |
| 61 | \( 1 - 1.01T + 61T^{2} \) |
| 67 | \( 1 + 5.11T + 67T^{2} \) |
| 71 | \( 1 + 1.17T + 71T^{2} \) |
| 73 | \( 1 + 2.92T + 73T^{2} \) |
| 79 | \( 1 - 5.80T + 79T^{2} \) |
| 83 | \( 1 - 6.36T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−8.263100272129051290266097671998, −7.55131123704814875077445745486, −7.27371454182941754372173022185, −6.43014656575359221254274631231, −5.32713664519054628218904431087, −4.13430528609972097433952428872, −3.21664556058645865702798750777, −2.36177501368075252289866579609, −1.66993277110605408880239821248, −1.15108335271065277464010592776,
1.15108335271065277464010592776, 1.66993277110605408880239821248, 2.36177501368075252289866579609, 3.21664556058645865702798750777, 4.13430528609972097433952428872, 5.32713664519054628218904431087, 6.43014656575359221254274631231, 7.27371454182941754372173022185, 7.55131123704814875077445745486, 8.263100272129051290266097671998