L(s) = 1 | + 0.632·2-s + 1.44·3-s − 1.60·4-s + 3.81·5-s + 0.912·6-s + 5.06·7-s − 2.27·8-s − 0.915·9-s + 2.41·10-s + 0.203·11-s − 2.31·12-s − 0.0723·13-s + 3.20·14-s + 5.51·15-s + 1.76·16-s − 2.98·17-s − 0.578·18-s + 1.18·19-s − 6.10·20-s + 7.31·21-s + 0.128·22-s + 1.99·23-s − 3.28·24-s + 9.56·25-s − 0.0457·26-s − 5.65·27-s − 8.11·28-s + ⋯ |
L(s) = 1 | + 0.446·2-s + 0.833·3-s − 0.800·4-s + 1.70·5-s + 0.372·6-s + 1.91·7-s − 0.804·8-s − 0.305·9-s + 0.762·10-s + 0.0613·11-s − 0.667·12-s − 0.0200·13-s + 0.856·14-s + 1.42·15-s + 0.440·16-s − 0.725·17-s − 0.136·18-s + 0.271·19-s − 1.36·20-s + 1.59·21-s + 0.0274·22-s + 0.415·23-s − 0.670·24-s + 1.91·25-s − 0.00897·26-s − 1.08·27-s − 1.53·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.044420519\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.044420519\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2671 | \( 1 - T \) |
good | 2 | \( 1 - 0.632T + 2T^{2} \) |
| 3 | \( 1 - 1.44T + 3T^{2} \) |
| 5 | \( 1 - 3.81T + 5T^{2} \) |
| 7 | \( 1 - 5.06T + 7T^{2} \) |
| 11 | \( 1 - 0.203T + 11T^{2} \) |
| 13 | \( 1 + 0.0723T + 13T^{2} \) |
| 17 | \( 1 + 2.98T + 17T^{2} \) |
| 19 | \( 1 - 1.18T + 19T^{2} \) |
| 23 | \( 1 - 1.99T + 23T^{2} \) |
| 29 | \( 1 - 4.82T + 29T^{2} \) |
| 31 | \( 1 + 6.41T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 - 0.854T + 41T^{2} \) |
| 43 | \( 1 - 1.93T + 43T^{2} \) |
| 47 | \( 1 + 6.53T + 47T^{2} \) |
| 53 | \( 1 + 9.85T + 53T^{2} \) |
| 59 | \( 1 - 6.25T + 59T^{2} \) |
| 61 | \( 1 + 5.00T + 61T^{2} \) |
| 67 | \( 1 + 3.02T + 67T^{2} \) |
| 71 | \( 1 - 2.95T + 71T^{2} \) |
| 73 | \( 1 + 16.4T + 73T^{2} \) |
| 79 | \( 1 + 4.77T + 79T^{2} \) |
| 83 | \( 1 - 4.76T + 83T^{2} \) |
| 89 | \( 1 - 1.03T + 89T^{2} \) |
| 97 | \( 1 + 5.45T + 97T^{2} \) |
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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
−8.918289712814888083577513511772, −8.296406397767568252593853856797, −7.55723039565576147849107471699, −6.26253579154702969222419450944, −5.57861824284944422519409013285, −4.93008818151033703535358008916, −4.29604642007813097004661049150, −2.97836568744122052766170082754, −2.19621244417251539876022511725, −1.30761764995234486195321092146,
1.30761764995234486195321092146, 2.19621244417251539876022511725, 2.97836568744122052766170082754, 4.29604642007813097004661049150, 4.93008818151033703535358008916, 5.57861824284944422519409013285, 6.26253579154702969222419450944, 7.55723039565576147849107471699, 8.296406397767568252593853856797, 8.918289712814888083577513511772