| L(s) = 1 | + 2.41·2-s + 2·3-s + 3.82·4-s + 5-s + 4.82·6-s + 0.828·7-s + 4.41·8-s + 9-s + 2.41·10-s + 4.82·11-s + 7.65·12-s − 2·13-s + 1.99·14-s + 2·15-s + 2.99·16-s + 2.82·17-s + 2.41·18-s − 0.828·19-s + 3.82·20-s + 1.65·21-s + 11.6·22-s − 8.82·23-s + 8.82·24-s + 25-s − 4.82·26-s − 4·27-s + 3.17·28-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 1.15·3-s + 1.91·4-s + 0.447·5-s + 1.97·6-s + 0.313·7-s + 1.56·8-s + 0.333·9-s + 0.763·10-s + 1.45·11-s + 2.21·12-s − 0.554·13-s + 0.534·14-s + 0.516·15-s + 0.749·16-s + 0.685·17-s + 0.569·18-s − 0.190·19-s + 0.856·20-s + 0.361·21-s + 2.48·22-s − 1.84·23-s + 1.80·24-s + 0.200·25-s − 0.946·26-s − 0.769·27-s + 0.599·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.335969482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.335969482\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 3 | \( 1 - 2T + 3T^{2} \) |
| 7 | \( 1 - 0.828T + 7T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 + 0.828T + 19T^{2} \) |
| 23 | \( 1 + 8.82T + 23T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 - 0.343T + 47T^{2} \) |
| 53 | \( 1 - 7.65T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 7.65T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 - 7.31T + 71T^{2} \) |
| 73 | \( 1 - 8.48T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + 3.65T + 89T^{2} \) |
| 97 | \( 1 + 4.48T + 97T^{2} \) |
| 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
−8.260980620920110319356907627697, −7.58968622228679854777188072722, −6.67715121169553801188843580131, −6.09378163345246889138794663444, −5.37874439054670968513073398139, −4.34944282207797482252890903747, −3.93601496109771530976076844967, −3.06977202321369680576306139710, −2.35046230384966630088696939815, −1.55606444348503665145752333763,
1.55606444348503665145752333763, 2.35046230384966630088696939815, 3.06977202321369680576306139710, 3.93601496109771530976076844967, 4.34944282207797482252890903747, 5.37874439054670968513073398139, 6.09378163345246889138794663444, 6.67715121169553801188843580131, 7.58968622228679854777188072722, 8.260980620920110319356907627697
