L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 2·7-s − 8-s + 9-s − 0.864·11-s + 12-s + 5.52·13-s − 2·14-s + 16-s − 3.52·17-s − 18-s + 8.11·19-s + 2·21-s + 0.864·22-s + 23-s − 24-s − 5.52·26-s + 27-s + 2·28-s − 2·29-s − 3.25·31-s − 32-s − 0.864·33-s + 3.52·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 0.333·9-s − 0.260·11-s + 0.288·12-s + 1.53·13-s − 0.534·14-s + 0.250·16-s − 0.854·17-s − 0.235·18-s + 1.86·19-s + 0.436·21-s + 0.184·22-s + 0.208·23-s − 0.204·24-s − 1.08·26-s + 0.192·27-s + 0.377·28-s − 0.371·29-s − 0.584·31-s − 0.176·32-s − 0.150·33-s + 0.604·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.076411923\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.076411923\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 0.864T + 11T^{2} \) |
| 13 | \( 1 - 5.52T + 13T^{2} \) |
| 17 | \( 1 + 3.52T + 17T^{2} \) |
| 19 | \( 1 - 8.11T + 19T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 3.25T + 31T^{2} \) |
| 37 | \( 1 - 5.13T + 37T^{2} \) |
| 41 | \( 1 - 3.52T + 41T^{2} \) |
| 43 | \( 1 - 1.13T + 43T^{2} \) |
| 47 | \( 1 - 5.52T + 47T^{2} \) |
| 53 | \( 1 - 1.34T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 + 5.91T + 61T^{2} \) |
| 67 | \( 1 + 5.91T + 67T^{2} \) |
| 71 | \( 1 + 5.79T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 + 3.79T + 79T^{2} \) |
| 83 | \( 1 - 8.32T + 83T^{2} \) |
| 89 | \( 1 - 4.77T + 89T^{2} \) |
| 97 | \( 1 + 1.04T + 97T^{2} \) |
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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
−8.744143814609026326969450312835, −7.73116404826478698795882735062, −7.57900393836366930215123722524, −6.46086369206727259843731827187, −5.69833334284756150834299490358, −4.74689428183184948008938247425, −3.74327398602902978128976013425, −2.92504896486029254582775517914, −1.84046435409098792438636235853, −0.992230543787200715461200273952,
0.992230543787200715461200273952, 1.84046435409098792438636235853, 2.92504896486029254582775517914, 3.74327398602902978128976013425, 4.74689428183184948008938247425, 5.69833334284756150834299490358, 6.46086369206727259843731827187, 7.57900393836366930215123722524, 7.73116404826478698795882735062, 8.744143814609026326969450312835