| L(s) = 1 | − 2-s − 1.21·3-s + 4-s + 1.21·6-s + 3.59·7-s − 8-s − 1.52·9-s − 4.73·11-s − 1.21·12-s + 1.78·13-s − 3.59·14-s + 16-s + 1.83·17-s + 1.52·18-s + 3.28·19-s − 4.36·21-s + 4.73·22-s − 1.80·23-s + 1.21·24-s − 1.78·26-s + 5.49·27-s + 3.59·28-s − 0.755·29-s − 2.06·31-s − 32-s + 5.75·33-s − 1.83·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.701·3-s + 0.5·4-s + 0.495·6-s + 1.35·7-s − 0.353·8-s − 0.508·9-s − 1.42·11-s − 0.350·12-s + 0.495·13-s − 0.960·14-s + 0.250·16-s + 0.445·17-s + 0.359·18-s + 0.752·19-s − 0.951·21-s + 1.01·22-s − 0.376·23-s + 0.247·24-s − 0.350·26-s + 1.05·27-s + 0.678·28-s − 0.140·29-s − 0.371·31-s − 0.176·32-s + 1.00·33-s − 0.314·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9411390628\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9411390628\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 + 1.21T + 3T^{2} \) |
| 7 | \( 1 - 3.59T + 7T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 13 | \( 1 - 1.78T + 13T^{2} \) |
| 17 | \( 1 - 1.83T + 17T^{2} \) |
| 19 | \( 1 - 3.28T + 19T^{2} \) |
| 23 | \( 1 + 1.80T + 23T^{2} \) |
| 29 | \( 1 + 0.755T + 29T^{2} \) |
| 31 | \( 1 + 2.06T + 31T^{2} \) |
| 41 | \( 1 + 2.57T + 41T^{2} \) |
| 43 | \( 1 - 9.19T + 43T^{2} \) |
| 47 | \( 1 - 1.24T + 47T^{2} \) |
| 53 | \( 1 - 6.56T + 53T^{2} \) |
| 59 | \( 1 - 9.19T + 59T^{2} \) |
| 61 | \( 1 + 9.39T + 61T^{2} \) |
| 67 | \( 1 + 2.39T + 67T^{2} \) |
| 71 | \( 1 + 9.39T + 71T^{2} \) |
| 73 | \( 1 - 8.09T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 + 0.933T + 89T^{2} \) |
| 97 | \( 1 + 2.42T + 97T^{2} \) |
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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
−9.151277243766888623845094382137, −8.307383448321807694381800379331, −7.85036462839381792939398339420, −7.09754322061188572025968622186, −5.77614325048529465289999736294, −5.48792846233368296067732696683, −4.53128558814086811483978676797, −3.10225356334277528534008017688, −2.02166940071156190352713018972, −0.75856436072018037282340273786,
0.75856436072018037282340273786, 2.02166940071156190352713018972, 3.10225356334277528534008017688, 4.53128558814086811483978676797, 5.48792846233368296067732696683, 5.77614325048529465289999736294, 7.09754322061188572025968622186, 7.85036462839381792939398339420, 8.307383448321807694381800379331, 9.151277243766888623845094382137
