L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 2.37·7-s − 8-s + 9-s − 10-s + 4.37·11-s − 12-s − 4.37·13-s − 2.37·14-s − 15-s + 16-s + 4.37·17-s − 18-s + 4.37·19-s + 20-s − 2.37·21-s − 4.37·22-s − 2.37·23-s + 24-s + 25-s + 4.37·26-s − 27-s + 2.37·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 0.896·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 1.31·11-s − 0.288·12-s − 1.21·13-s − 0.634·14-s − 0.258·15-s + 0.250·16-s + 1.06·17-s − 0.235·18-s + 1.00·19-s + 0.223·20-s − 0.517·21-s − 0.932·22-s − 0.494·23-s + 0.204·24-s + 0.200·25-s + 0.857·26-s − 0.192·27-s + 0.448·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.247897075\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.247897075\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 37 | \( 1 + T \) |
good | 7 | \( 1 - 2.37T + 7T^{2} \) |
| 11 | \( 1 - 4.37T + 11T^{2} \) |
| 13 | \( 1 + 4.37T + 13T^{2} \) |
| 17 | \( 1 - 4.37T + 17T^{2} \) |
| 19 | \( 1 - 4.37T + 19T^{2} \) |
| 23 | \( 1 + 2.37T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 8.74T + 31T^{2} \) |
| 41 | \( 1 + 2.74T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 6.74T + 47T^{2} \) |
| 53 | \( 1 + 0.372T + 53T^{2} \) |
| 59 | \( 1 - 3.25T + 59T^{2} \) |
| 61 | \( 1 - 8.74T + 61T^{2} \) |
| 67 | \( 1 + 8.74T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 6.37T + 83T^{2} \) |
| 89 | \( 1 - 7.62T + 89T^{2} \) |
| 97 | \( 1 + 9.48T + 97T^{2} \) |
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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
−9.785166473869818352386352040370, −9.233962082308994120919370484186, −8.166292991905687590382797175022, −7.37109103034187218877082252533, −6.64367677864087529475494925552, −5.58163714663183829670662499015, −4.89680318143745927342182143807, −3.57733667604527650710703646354, −2.06079048662367972003274439679, −1.04556555540668435101019840246,
1.04556555540668435101019840246, 2.06079048662367972003274439679, 3.57733667604527650710703646354, 4.89680318143745927342182143807, 5.58163714663183829670662499015, 6.64367677864087529475494925552, 7.37109103034187218877082252533, 8.166292991905687590382797175022, 9.233962082308994120919370484186, 9.785166473869818352386352040370