L(s) = 1 | + 2-s + 4-s + 2·5-s + 7-s + 8-s + 2·10-s − 6·13-s + 14-s + 16-s + 2·17-s + 4·19-s + 2·20-s − 8·23-s − 25-s − 6·26-s + 28-s − 2·29-s + 32-s + 2·34-s + 2·35-s − 10·37-s + 4·38-s + 2·40-s − 6·41-s + 4·43-s − 8·46-s + 49-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.377·7-s + 0.353·8-s + 0.632·10-s − 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.917·19-s + 0.447·20-s − 1.66·23-s − 1/5·25-s − 1.17·26-s + 0.188·28-s − 0.371·29-s + 0.176·32-s + 0.342·34-s + 0.338·35-s − 1.64·37-s + 0.648·38-s + 0.316·40-s − 0.937·41-s + 0.609·43-s − 1.17·46-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15246 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15246 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−16.31267513124325, −15.60445868031024, −15.17032704235407, −14.35530197601081, −14.09377751769397, −13.78729338684191, −12.95942113442203, −12.37901252685784, −11.90725735401329, −11.53079871542511, −10.48838121795106, −10.12333041188748, −9.654745527110712, −8.990380449699339, −8.041794720972076, −7.530512977757508, −7.009640766623148, −6.128652808358110, −5.620417450437192, −5.089025169518979, −4.491019104974876, −3.604559496147143, −2.849122467732106, −2.046103710602013, −1.549709282539988, 0,
1.549709282539988, 2.046103710602013, 2.849122467732106, 3.604559496147143, 4.491019104974876, 5.089025169518979, 5.620417450437192, 6.128652808358110, 7.009640766623148, 7.530512977757508, 8.041794720972076, 8.990380449699339, 9.654745527110712, 10.12333041188748, 10.48838121795106, 11.53079871542511, 11.90725735401329, 12.37901252685784, 12.95942113442203, 13.78729338684191, 14.09377751769397, 14.35530197601081, 15.17032704235407, 15.60445868031024, 16.31267513124325