| L(s) = 1 | + 2.73·2-s − 3-s + 5.46·4-s − 0.332·5-s − 2.73·6-s − 7-s + 9.48·8-s + 9-s − 0.908·10-s + 3.68·11-s − 5.46·12-s − 1.12·13-s − 2.73·14-s + 0.332·15-s + 14.9·16-s − 6.80·17-s + 2.73·18-s + 7.98·19-s − 1.81·20-s + 21-s + 10.0·22-s + 5.50·23-s − 9.48·24-s − 4.88·25-s − 3.06·26-s − 27-s − 5.46·28-s + ⋯ |
| L(s) = 1 | + 1.93·2-s − 0.577·3-s + 2.73·4-s − 0.148·5-s − 1.11·6-s − 0.377·7-s + 3.35·8-s + 0.333·9-s − 0.287·10-s + 1.10·11-s − 1.57·12-s − 0.311·13-s − 0.730·14-s + 0.0857·15-s + 3.74·16-s − 1.65·17-s + 0.644·18-s + 1.83·19-s − 0.406·20-s + 0.218·21-s + 2.14·22-s + 1.14·23-s − 1.93·24-s − 0.977·25-s − 0.601·26-s − 0.192·27-s − 1.03·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.069053119\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.069053119\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 - T \) |
| good | 2 | \( 1 - 2.73T + 2T^{2} \) |
| 5 | \( 1 + 0.332T + 5T^{2} \) |
| 11 | \( 1 - 3.68T + 11T^{2} \) |
| 13 | \( 1 + 1.12T + 13T^{2} \) |
| 17 | \( 1 + 6.80T + 17T^{2} \) |
| 19 | \( 1 - 7.98T + 19T^{2} \) |
| 23 | \( 1 - 5.50T + 23T^{2} \) |
| 29 | \( 1 - 7.37T + 29T^{2} \) |
| 31 | \( 1 - 5.70T + 31T^{2} \) |
| 37 | \( 1 + 7.48T + 37T^{2} \) |
| 41 | \( 1 + 2.47T + 41T^{2} \) |
| 43 | \( 1 + 1.28T + 43T^{2} \) |
| 47 | \( 1 + 4.02T + 47T^{2} \) |
| 53 | \( 1 + 6.23T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 - 16.0T + 73T^{2} \) |
| 79 | \( 1 - 7.62T + 79T^{2} \) |
| 83 | \( 1 - 0.0940T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−8.154549612538666926186285754741, −7.11939065987652370398927626846, −6.67519233159782455283542265726, −6.26341489714662244494610988610, −5.15315236379121458918080850603, −4.85802469241998107378818503230, −3.92514395502338565552963168505, −3.29310629561019461797745677272, −2.33818085796716034320262687202, −1.17965247281596016873202568516,
1.17965247281596016873202568516, 2.33818085796716034320262687202, 3.29310629561019461797745677272, 3.92514395502338565552963168505, 4.85802469241998107378818503230, 5.15315236379121458918080850603, 6.26341489714662244494610988610, 6.67519233159782455283542265726, 7.11939065987652370398927626846, 8.154549612538666926186285754741
