| L(s) = 1 | + 2-s + 1.66·3-s + 4-s − 3.73·5-s + 1.66·6-s − 7-s + 8-s − 0.226·9-s − 3.73·10-s + 4.90·11-s + 1.66·12-s − 3.86·13-s − 14-s − 6.21·15-s + 16-s + 1.41·17-s − 0.226·18-s + 7.06·19-s − 3.73·20-s − 1.66·21-s + 4.90·22-s − 6.30·23-s + 1.66·24-s + 8.92·25-s − 3.86·26-s − 5.37·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.961·3-s + 0.5·4-s − 1.66·5-s + 0.679·6-s − 0.377·7-s + 0.353·8-s − 0.0753·9-s − 1.18·10-s + 1.47·11-s + 0.480·12-s − 1.07·13-s − 0.267·14-s − 1.60·15-s + 0.250·16-s + 0.344·17-s − 0.0532·18-s + 1.62·19-s − 0.834·20-s − 0.363·21-s + 1.04·22-s − 1.31·23-s + 0.339·24-s + 1.78·25-s − 0.758·26-s − 1.03·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.035654098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.035654098\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 - 1.66T + 3T^{2} \) |
| 5 | \( 1 + 3.73T + 5T^{2} \) |
| 11 | \( 1 - 4.90T + 11T^{2} \) |
| 13 | \( 1 + 3.86T + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 - 7.06T + 19T^{2} \) |
| 23 | \( 1 + 6.30T + 23T^{2} \) |
| 29 | \( 1 + 3.28T + 29T^{2} \) |
| 31 | \( 1 - 6.86T + 31T^{2} \) |
| 37 | \( 1 + 6.09T + 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 + 8.07T + 43T^{2} \) |
| 47 | \( 1 - 1.41T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 + 2.12T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 5.66T + 71T^{2} \) |
| 73 | \( 1 - 7.55T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 + 5.21T + 83T^{2} \) |
| 89 | \( 1 + 6.34T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−8.074852541917807095057978515965, −7.28163688784394198763167515614, −6.97148366086779789037434895310, −5.89791885780561998201575918176, −5.02459622088959451804356896748, −4.04396166388472223480005643174, −3.72285049984168921683559690265, −3.10417176124116478671136899076, −2.20063204953476730480412724391, −0.77440931710387553153039996219,
0.77440931710387553153039996219, 2.20063204953476730480412724391, 3.10417176124116478671136899076, 3.72285049984168921683559690265, 4.04396166388472223480005643174, 5.02459622088959451804356896748, 5.89791885780561998201575918176, 6.97148366086779789037434895310, 7.28163688784394198763167515614, 8.074852541917807095057978515965
