| L(s) = 1 | + 2-s + 4-s − 5-s + 3·7-s + 8-s − 10-s + 11-s − 3.81·13-s + 3·14-s + 16-s − 17-s + 7.81·19-s − 20-s + 22-s − 5.81·23-s + 25-s − 3.81·26-s + 3·28-s + 6.81·29-s + 6.81·31-s + 32-s − 34-s − 3·35-s + 37-s + 7.81·38-s − 40-s + 4.81·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s + 1.13·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s − 1.05·13-s + 0.801·14-s + 0.250·16-s − 0.242·17-s + 1.79·19-s − 0.223·20-s + 0.213·22-s − 1.21·23-s + 0.200·25-s − 0.748·26-s + 0.566·28-s + 1.26·29-s + 1.22·31-s + 0.176·32-s − 0.171·34-s − 0.507·35-s + 0.164·37-s + 1.26·38-s − 0.158·40-s + 0.751·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.233644279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.233644279\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + 3.81T + 13T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 19 | \( 1 - 7.81T + 19T^{2} \) |
| 23 | \( 1 + 5.81T + 23T^{2} \) |
| 29 | \( 1 - 6.81T + 29T^{2} \) |
| 31 | \( 1 - 6.81T + 31T^{2} \) |
| 41 | \( 1 - 4.81T + 41T^{2} \) |
| 43 | \( 1 + 4.81T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 + 2T + 59T^{2} \) |
| 61 | \( 1 - 6.81T + 61T^{2} \) |
| 67 | \( 1 - 7.63T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 5.81T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 - 3.81T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + 4.81T + 97T^{2} \) |
| 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
−8.280523531814615057916924495189, −7.888925148562144736579806790947, −7.16258266562957154872810420098, −6.35167059059552967495507606430, −5.34572518032731115628669759423, −4.78173537074887933805717113272, −4.13684221576264891457456774685, −3.09103545793577041423842447186, −2.19556230189294921149445801233, −1.01511812341425361058763627012,
1.01511812341425361058763627012, 2.19556230189294921149445801233, 3.09103545793577041423842447186, 4.13684221576264891457456774685, 4.78173537074887933805717113272, 5.34572518032731115628669759423, 6.35167059059552967495507606430, 7.16258266562957154872810420098, 7.888925148562144736579806790947, 8.280523531814615057916924495189
