| L(s) = 1 | + 4.81·2-s + 3·3-s + 15.1·4-s + 15.3·5-s + 14.4·6-s + 25.8·7-s + 34.4·8-s + 9·9-s + 74.0·10-s + 24.8·11-s + 45.5·12-s + 6.10·13-s + 124.·14-s + 46.1·15-s + 44.7·16-s − 129.·17-s + 43.3·18-s + 135.·19-s + 233.·20-s + 77.4·21-s + 119.·22-s + 23·23-s + 103.·24-s + 111.·25-s + 29.3·26-s + 27·27-s + 391.·28-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 0.577·3-s + 1.89·4-s + 1.37·5-s + 0.982·6-s + 1.39·7-s + 1.52·8-s + 0.333·9-s + 2.34·10-s + 0.681·11-s + 1.09·12-s + 0.130·13-s + 2.37·14-s + 0.794·15-s + 0.698·16-s − 1.84·17-s + 0.567·18-s + 1.63·19-s + 2.60·20-s + 0.805·21-s + 1.15·22-s + 0.208·23-s + 0.880·24-s + 0.891·25-s + 0.221·26-s + 0.192·27-s + 2.64·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(12.91021699\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.91021699\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 23 | \( 1 - 23T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 4.81T + 8T^{2} \) |
| 5 | \( 1 - 15.3T + 125T^{2} \) |
| 7 | \( 1 - 25.8T + 343T^{2} \) |
| 11 | \( 1 - 24.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 6.10T + 2.19e3T^{2} \) |
| 17 | \( 1 + 129.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 135.T + 6.85e3T^{2} \) |
| 31 | \( 1 + 50.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 402.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 119.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 218.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 491.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 657.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 205.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 486.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 300.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.02e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 816.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 160.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.08e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.33e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.09e3T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.903148217321239901068522979263, −7.84996943607123728858151018810, −6.83654212746128679710097229639, −6.33700939740684329007572339950, −5.17086946695198804006695309154, −5.00268166344705679059115734041, −3.95107053748134927734679643731, −3.00569411482521911213051896727, −1.94751309925293220534755948247, −1.59490032826673218518610552887,
1.59490032826673218518610552887, 1.94751309925293220534755948247, 3.00569411482521911213051896727, 3.95107053748134927734679643731, 5.00268166344705679059115734041, 5.17086946695198804006695309154, 6.33700939740684329007572339950, 6.83654212746128679710097229639, 7.84996943607123728858151018810, 8.903148217321239901068522979263
