L(s) = 1 | − 2-s − 1.35·3-s + 4-s + 1.35·5-s + 1.35·6-s − 0.960·7-s − 8-s − 1.16·9-s − 1.35·10-s + 2.31·11-s − 1.35·12-s + 0.313·13-s + 0.960·14-s − 1.83·15-s + 16-s − 2.84·17-s + 1.16·18-s + 0.816·19-s + 1.35·20-s + 1.29·21-s − 2.31·22-s − 23-s + 1.35·24-s − 3.16·25-s − 0.313·26-s + 5.64·27-s − 0.960·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.781·3-s + 0.5·4-s + 0.605·5-s + 0.552·6-s − 0.363·7-s − 0.353·8-s − 0.389·9-s − 0.427·10-s + 0.697·11-s − 0.390·12-s + 0.0869·13-s + 0.256·14-s − 0.472·15-s + 0.250·16-s − 0.691·17-s + 0.275·18-s + 0.187·19-s + 0.302·20-s + 0.283·21-s − 0.493·22-s − 0.208·23-s + 0.276·24-s − 0.633·25-s − 0.0615·26-s + 1.08·27-s − 0.181·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
good | 3 | \( 1 + 1.35T + 3T^{2} \) |
| 5 | \( 1 - 1.35T + 5T^{2} \) |
| 7 | \( 1 + 0.960T + 7T^{2} \) |
| 11 | \( 1 - 2.31T + 11T^{2} \) |
| 13 | \( 1 - 0.313T + 13T^{2} \) |
| 17 | \( 1 + 2.84T + 17T^{2} \) |
| 19 | \( 1 - 0.816T + 19T^{2} \) |
| 31 | \( 1 + 5.68T + 31T^{2} \) |
| 37 | \( 1 - 6.29T + 37T^{2} \) |
| 41 | \( 1 + 7.58T + 41T^{2} \) |
| 43 | \( 1 - 3.94T + 43T^{2} \) |
| 47 | \( 1 + 1.24T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + 4.62T + 59T^{2} \) |
| 61 | \( 1 + 6.96T + 61T^{2} \) |
| 67 | \( 1 - 0.849T + 67T^{2} \) |
| 71 | \( 1 + 4.33T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 + 3.01T + 79T^{2} \) |
| 83 | \( 1 - 0.366T + 83T^{2} \) |
| 89 | \( 1 + 5.21T + 89T^{2} \) |
| 97 | \( 1 + 5.41T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.249087586088838926819576859971, −8.636257253569641130430757731633, −7.54655838570794790981591910358, −6.60121496703643990231141691070, −6.06144937696711355189013464842, −5.29869615302320818252843018020, −4.01424333125866348756512151561, −2.73545943910343702383571569010, −1.51934110215646368009139690093, 0,
1.51934110215646368009139690093, 2.73545943910343702383571569010, 4.01424333125866348756512151561, 5.29869615302320818252843018020, 6.06144937696711355189013464842, 6.60121496703643990231141691070, 7.54655838570794790981591910358, 8.636257253569641130430757731633, 9.249087586088838926819576859971