L(s) = 1 | + 1.36·2-s − 0.129·4-s + 0.636·5-s + 7-s − 2.91·8-s + 0.870·10-s − 3.59·13-s + 1.36·14-s − 3.72·16-s + 1.46·17-s − 5.85·19-s − 0.0824·20-s + 8.56·23-s − 4.59·25-s − 4.91·26-s − 0.129·28-s + 7.73·29-s − 9.44·31-s + 0.731·32-s + 2·34-s + 0.636·35-s + 5.59·37-s − 8.00·38-s − 1.85·40-s + 8.37·41-s + 1.74·43-s + 11.7·46-s + ⋯ |
L(s) = 1 | + 0.967·2-s − 0.0647·4-s + 0.284·5-s + 0.377·7-s − 1.02·8-s + 0.275·10-s − 0.997·13-s + 0.365·14-s − 0.931·16-s + 0.354·17-s − 1.34·19-s − 0.0184·20-s + 1.78·23-s − 0.918·25-s − 0.964·26-s − 0.0244·28-s + 1.43·29-s − 1.69·31-s + 0.129·32-s + 0.342·34-s + 0.107·35-s + 0.919·37-s − 1.29·38-s − 0.293·40-s + 1.30·41-s + 0.265·43-s + 1.72·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.695045761\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.695045761\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 1.36T + 2T^{2} \) |
| 5 | \( 1 - 0.636T + 5T^{2} \) |
| 13 | \( 1 + 3.59T + 13T^{2} \) |
| 17 | \( 1 - 1.46T + 17T^{2} \) |
| 19 | \( 1 + 5.85T + 19T^{2} \) |
| 23 | \( 1 - 8.56T + 23T^{2} \) |
| 29 | \( 1 - 7.73T + 29T^{2} \) |
| 31 | \( 1 + 9.44T + 31T^{2} \) |
| 37 | \( 1 - 5.59T + 37T^{2} \) |
| 41 | \( 1 - 8.37T + 41T^{2} \) |
| 43 | \( 1 - 1.74T + 43T^{2} \) |
| 47 | \( 1 + 4.99T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 1.85T + 67T^{2} \) |
| 71 | \( 1 - 9.66T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 4.51T + 79T^{2} \) |
| 83 | \( 1 - 3.08T + 83T^{2} \) |
| 89 | \( 1 - 4.19T + 89T^{2} \) |
| 97 | \( 1 - 7.44T + 97T^{2} \) |
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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
−7.82116225243411739036404377428, −7.01842508533187378312603539545, −6.33567660970279767286833926937, −5.59242548594641137501371642051, −5.00061506095793156474986931483, −4.43479820599443244861896516794, −3.69675929417969286816618255226, −2.75870396242534950895593793844, −2.11361236339331091022617459059, −0.69309319429416206093615435197,
0.69309319429416206093615435197, 2.11361236339331091022617459059, 2.75870396242534950895593793844, 3.69675929417969286816618255226, 4.43479820599443244861896516794, 5.00061506095793156474986931483, 5.59242548594641137501371642051, 6.33567660970279767286833926937, 7.01842508533187378312603539545, 7.82116225243411739036404377428