L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 0.723·7-s + 8-s + 9-s + 5.51·11-s − 12-s − 4.96·13-s − 0.723·14-s + 16-s − 4.23·17-s + 18-s + 4.96·19-s + 0.723·21-s + 5.51·22-s + 23-s − 24-s − 4.96·26-s − 27-s − 0.723·28-s − 29-s + 6·31-s + 32-s − 5.51·33-s − 4.23·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s − 0.273·7-s + 0.353·8-s + 0.333·9-s + 1.66·11-s − 0.288·12-s − 1.37·13-s − 0.193·14-s + 0.250·16-s − 1.02·17-s + 0.235·18-s + 1.13·19-s + 0.157·21-s + 1.17·22-s + 0.208·23-s − 0.204·24-s − 0.973·26-s − 0.192·27-s − 0.136·28-s − 0.185·29-s + 1.07·31-s + 0.176·32-s − 0.959·33-s − 0.726·34-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
\(L(1)\) |
\(\approx\) |
\(2.481913425\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.481913425\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.723T + 7T^{2} \) |
| 11 | \( 1 - 5.51T + 11T^{2} \) |
| 13 | \( 1 + 4.96T + 13T^{2} \) |
| 17 | \( 1 + 4.23T + 17T^{2} \) |
| 19 | \( 1 - 4.96T + 19T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 4.23T + 37T^{2} \) |
| 41 | \( 1 - 1.03T + 41T^{2} \) |
| 43 | \( 1 - 0.961T + 43T^{2} \) |
| 47 | \( 1 - 7.96T + 47T^{2} \) |
| 53 | \( 1 - 6.47T + 53T^{2} \) |
| 59 | \( 1 + 9.02T + 59T^{2} \) |
| 61 | \( 1 - 7.44T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 + 0.514T + 71T^{2} \) |
| 73 | \( 1 - 0.447T + 73T^{2} \) |
| 79 | \( 1 + 5.51T + 79T^{2} \) |
| 83 | \( 1 + 6.17T + 83T^{2} \) |
| 89 | \( 1 + 2.23T + 89T^{2} \) |
| 97 | \( 1 + 2.55T + 97T^{2} \) |
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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
−8.646138932500014988029571518426, −7.49056218648307220646253640934, −6.93668978321231570546409782779, −6.34776260122142337732131942044, −5.56451027766181819697156160877, −4.67583871533709853650600998050, −4.15896441470972174861337425961, −3.13517566664767729071453204154, −2.12262401440604801081396194252, −0.880356685132832963589608933377,
0.880356685132832963589608933377, 2.12262401440604801081396194252, 3.13517566664767729071453204154, 4.15896441470972174861337425961, 4.67583871533709853650600998050, 5.56451027766181819697156160877, 6.34776260122142337732131942044, 6.93668978321231570546409782779, 7.49056218648307220646253640934, 8.646138932500014988029571518426