L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 3.12·7-s − 8-s + 9-s − 10-s − 5.12·11-s + 12-s + 3.12·14-s + 15-s + 16-s − 2·17-s − 18-s − 6·19-s + 20-s − 3.12·21-s + 5.12·22-s + 5.12·23-s − 24-s + 25-s + 27-s − 3.12·28-s + 2·29-s − 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 1.18·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 1.54·11-s + 0.288·12-s + 0.834·14-s + 0.258·15-s + 0.250·16-s − 0.485·17-s − 0.235·18-s − 1.37·19-s + 0.223·20-s − 0.681·21-s + 1.09·22-s + 1.06·23-s − 0.204·24-s + 0.200·25-s + 0.192·27-s − 0.590·28-s + 0.371·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.130179314\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.130179314\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + 3.12T + 7T^{2} \) |
| 11 | \( 1 + 5.12T + 11T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 5.12T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 3.12T + 31T^{2} \) |
| 37 | \( 1 - 5.12T + 37T^{2} \) |
| 41 | \( 1 - 0.876T + 41T^{2} \) |
| 43 | \( 1 + 6.24T + 43T^{2} \) |
| 47 | \( 1 - 6.24T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 - 1.12T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 4.87T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 6.24T + 83T^{2} \) |
| 89 | \( 1 + 3.12T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−8.324879362536268305226009075557, −7.68465894745741975349169286599, −6.71162115505676129564793190841, −6.43183015426782933376632376441, −5.40504721912048710630751567545, −4.54507390340573729070352715934, −3.38371555215605813375835115957, −2.67260669562618553857035774741, −2.10474281555997647732527398179, −0.59018947487253063624918640148,
0.59018947487253063624918640148, 2.10474281555997647732527398179, 2.67260669562618553857035774741, 3.38371555215605813375835115957, 4.54507390340573729070352715934, 5.40504721912048710630751567545, 6.43183015426782933376632376441, 6.71162115505676129564793190841, 7.68465894745741975349169286599, 8.324879362536268305226009075557