L(s) = 1 | − 2-s − 4-s − 2·5-s − 7-s + 3·8-s − 3·9-s + 2·10-s + 11-s − 13-s + 14-s − 16-s − 2·17-s + 3·18-s − 4·19-s + 2·20-s − 22-s − 25-s + 26-s + 28-s + 6·29-s − 4·31-s − 5·32-s + 2·34-s + 2·35-s + 3·36-s + 6·37-s + 4·38-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.894·5-s − 0.377·7-s + 1.06·8-s − 9-s + 0.632·10-s + 0.301·11-s − 0.277·13-s + 0.267·14-s − 1/4·16-s − 0.485·17-s + 0.707·18-s − 0.917·19-s + 0.447·20-s − 0.213·22-s − 1/5·25-s + 0.196·26-s + 0.188·28-s + 1.11·29-s − 0.718·31-s − 0.883·32-s + 0.342·34-s + 0.338·35-s + 1/2·36-s + 0.986·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4784304887\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4784304887\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−9.751006238980819961821982151295, −9.093925352010553749015761126762, −8.347715086926499957264902495141, −7.78115594232506052995598147111, −6.76613462384288733833635328872, −5.71916010141537657333693279542, −4.51871905000729428774558961941, −3.81630106708447715099293181307, −2.46510746308353991254277247738, −0.58060517937373454680463896492,
0.58060517937373454680463896492, 2.46510746308353991254277247738, 3.81630106708447715099293181307, 4.51871905000729428774558961941, 5.71916010141537657333693279542, 6.76613462384288733833635328872, 7.78115594232506052995598147111, 8.347715086926499957264902495141, 9.093925352010553749015761126762, 9.751006238980819961821982151295