| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s − 0.957·5-s − 6·6-s + 7·7-s − 8·8-s + 9·9-s + 1.91·10-s + 42.6·11-s + 12·12-s + 32.1·13-s − 14·14-s − 2.87·15-s + 16·16-s + 10.7·17-s − 18·18-s − 19·19-s − 3.83·20-s + 21·21-s − 85.3·22-s + 190.·23-s − 24·24-s − 124.·25-s − 64.3·26-s + 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.0856·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.0605·10-s + 1.16·11-s + 0.288·12-s + 0.686·13-s − 0.267·14-s − 0.0494·15-s + 0.250·16-s + 0.153·17-s − 0.235·18-s − 0.229·19-s − 0.0428·20-s + 0.218·21-s − 0.827·22-s + 1.72·23-s − 0.204·24-s − 0.992·25-s − 0.485·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.181033518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.181033518\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 - 7T \) |
| 19 | \( 1 + 19T \) |
| good | 5 | \( 1 + 0.957T + 125T^{2} \) |
| 11 | \( 1 - 42.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 32.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 10.7T + 4.91e3T^{2} \) |
| 23 | \( 1 - 190.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 278.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 289.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 216.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 62.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 16.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 25.3T + 1.03e5T^{2} \) |
| 53 | \( 1 - 706.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 655.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 291.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 688.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 768.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 446.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 385.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 797.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 847.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 307.T + 9.12e5T^{2} \) |
| 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
−9.678820778460937826406315757001, −8.935343754246711706229124389904, −8.403136088230482956367986075108, −7.39366630218444024596163931752, −6.68057855419727622934605903963, −5.58412538624552236995527523398, −4.21103251170042970067008257325, −3.29828707429487773094928894508, −1.95052513693247050463386446901, −0.951356660095165530972049685145,
0.951356660095165530972049685145, 1.95052513693247050463386446901, 3.29828707429487773094928894508, 4.21103251170042970067008257325, 5.58412538624552236995527523398, 6.68057855419727622934605903963, 7.39366630218444024596163931752, 8.403136088230482956367986075108, 8.935343754246711706229124389904, 9.678820778460937826406315757001
