| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 3.76·7-s + 8-s + 9-s + 3.18·11-s − 12-s + 6.34·13-s + 3.76·14-s + 16-s + 2.58·17-s + 18-s − 6.34·19-s − 3.76·21-s + 3.18·22-s + 23-s − 24-s + 6.34·26-s − 27-s + 3.76·28-s − 29-s + 6·31-s + 32-s − 3.18·33-s + 2.58·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s + 1.42·7-s + 0.353·8-s + 0.333·9-s + 0.959·11-s − 0.288·12-s + 1.75·13-s + 1.00·14-s + 0.250·16-s + 0.625·17-s + 0.235·18-s − 1.45·19-s − 0.821·21-s + 0.678·22-s + 0.208·23-s − 0.204·24-s + 1.24·26-s − 0.192·27-s + 0.711·28-s − 0.185·29-s + 1.07·31-s + 0.176·32-s − 0.554·33-s + 0.442·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.498754676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.498754676\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 3.76T + 7T^{2} \) |
| 11 | \( 1 - 3.18T + 11T^{2} \) |
| 13 | \( 1 - 6.34T + 13T^{2} \) |
| 17 | \( 1 - 2.58T + 17T^{2} \) |
| 19 | \( 1 + 6.34T + 19T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 2.58T + 37T^{2} \) |
| 41 | \( 1 - 12.3T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 3.34T + 47T^{2} \) |
| 53 | \( 1 + 7.16T + 53T^{2} \) |
| 59 | \( 1 + 4.36T + 59T^{2} \) |
| 61 | \( 1 + 1.52T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 - 1.81T + 71T^{2} \) |
| 73 | \( 1 + 8.52T + 73T^{2} \) |
| 79 | \( 1 + 3.18T + 79T^{2} \) |
| 83 | \( 1 - 7.28T + 83T^{2} \) |
| 89 | \( 1 - 4.58T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
| 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
−8.400266502739517266935623207203, −7.924309787287163073502306063216, −6.82765615799909854890119962310, −6.20965672461252008509097672160, −5.64129551076742281861215888922, −4.59522981110592240821265362233, −4.22772847930019121846716466753, −3.25081240524802601060802712696, −1.80365917695670907629311937473, −1.18666867206037629698864853977,
1.18666867206037629698864853977, 1.80365917695670907629311937473, 3.25081240524802601060802712696, 4.22772847930019121846716466753, 4.59522981110592240821265362233, 5.64129551076742281861215888922, 6.20965672461252008509097672160, 6.82765615799909854890119962310, 7.924309787287163073502306063216, 8.400266502739517266935623207203
