| L(s) = 1 | + 0.311·2-s − 1.90·4-s + 4.42·7-s − 1.21·8-s + 2.62·11-s − 0.474·13-s + 1.37·14-s + 3.42·16-s + 5.05·17-s − 19-s + 0.815·22-s − 1.37·23-s − 0.147·26-s − 8.42·28-s + 7.80·29-s + 1.24·31-s + 3.49·32-s + 1.57·34-s − 4.47·37-s − 0.311·38-s + 5.05·41-s − 12.0·43-s − 4.99·44-s − 0.428·46-s − 4.42·47-s + 12.6·49-s + 0.903·52-s + ⋯ |
| L(s) = 1 | + 0.219·2-s − 0.951·4-s + 1.67·7-s − 0.429·8-s + 0.790·11-s − 0.131·13-s + 0.368·14-s + 0.857·16-s + 1.22·17-s − 0.229·19-s + 0.173·22-s − 0.287·23-s − 0.0289·26-s − 1.59·28-s + 1.44·29-s + 0.223·31-s + 0.617·32-s + 0.269·34-s − 0.735·37-s − 0.0504·38-s + 0.788·41-s − 1.83·43-s − 0.752·44-s − 0.0631·46-s − 0.645·47-s + 1.80·49-s + 0.125·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.311282391\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.311282391\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 0.311T + 2T^{2} \) |
| 7 | \( 1 - 4.42T + 7T^{2} \) |
| 11 | \( 1 - 2.62T + 11T^{2} \) |
| 13 | \( 1 + 0.474T + 13T^{2} \) |
| 17 | \( 1 - 5.05T + 17T^{2} \) |
| 23 | \( 1 + 1.37T + 23T^{2} \) |
| 29 | \( 1 - 7.80T + 29T^{2} \) |
| 31 | \( 1 - 1.24T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 - 5.05T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 + 4.42T + 47T^{2} \) |
| 53 | \( 1 - 7.52T + 53T^{2} \) |
| 59 | \( 1 - 2.19T + 59T^{2} \) |
| 61 | \( 1 - 3.67T + 61T^{2} \) |
| 67 | \( 1 - 1.65T + 67T^{2} \) |
| 71 | \( 1 + 7.61T + 71T^{2} \) |
| 73 | \( 1 - 3.80T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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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
−8.445303815243107764094899877641, −7.87259017994606142883283081965, −7.00376502072586752291507398194, −5.97886510534836016244365900351, −5.23954147269880624012624165968, −4.66979218380673410910433879606, −4.02095876892330205371611994998, −3.09837715635027740627346052664, −1.75650834087892417656269647235, −0.912897243955345934612498192254,
0.912897243955345934612498192254, 1.75650834087892417656269647235, 3.09837715635027740627346052664, 4.02095876892330205371611994998, 4.66979218380673410910433879606, 5.23954147269880624012624165968, 5.97886510534836016244365900351, 7.00376502072586752291507398194, 7.87259017994606142883283081965, 8.445303815243107764094899877641
