L(s) = 1 | + 1.61·2-s − 1.61·3-s + 1.61·4-s − 2.61·6-s − 0.618·7-s + 8-s + 1.61·9-s − 11-s − 2.61·12-s − 13-s − 1.00·14-s + 2.61·18-s + 1.61·19-s + 1.00·21-s − 1.61·22-s + 0.618·23-s − 1.61·24-s + 25-s − 1.61·26-s − 27-s − 1.00·28-s − 32-s + 1.61·33-s + 2.61·36-s + 2.61·38-s + 1.61·39-s − 0.618·41-s + ⋯ |
L(s) = 1 | + 1.61·2-s − 1.61·3-s + 1.61·4-s − 2.61·6-s − 0.618·7-s + 8-s + 1.61·9-s − 11-s − 2.61·12-s − 13-s − 1.00·14-s + 2.61·18-s + 1.61·19-s + 1.00·21-s − 1.61·22-s + 0.618·23-s − 1.61·24-s + 25-s − 1.61·26-s − 27-s − 1.00·28-s − 32-s + 1.61·33-s + 2.61·36-s + 2.61·38-s + 1.61·39-s − 0.618·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8458447016\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8458447016\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 - 1.61T + T^{2} \) |
| 3 | \( 1 + 1.61T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + 0.618T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 1.61T + T^{2} \) |
| 23 | \( 1 - 0.618T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 0.618T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.61T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.61T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 0.618T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−13.01428957171829116532910775374, −12.49653956081239081415209781238, −11.67030199132299516852202375700, −10.82066324397992018341842346619, −9.724324737236978039854594585250, −7.34040097683464702793451134111, −6.44399786173049336138211999786, −5.31421961759558757154414127899, −4.86221255192137125487916254236, −3.07141793703332287184327843999,
3.07141793703332287184327843999, 4.86221255192137125487916254236, 5.31421961759558757154414127899, 6.44399786173049336138211999786, 7.34040097683464702793451134111, 9.724324737236978039854594585250, 10.82066324397992018341842346619, 11.67030199132299516852202375700, 12.49653956081239081415209781238, 13.01428957171829116532910775374