| L(s) = 1 | + 2·2-s + 5.96·3-s + 4·4-s + 11.9·6-s − 15.2·7-s + 8·8-s + 8.62·9-s + 12.0·11-s + 23.8·12-s − 90.1·13-s − 30.4·14-s + 16·16-s − 83.9·17-s + 17.2·18-s + 74.8·19-s − 91.0·21-s + 24.0·22-s − 23·23-s + 47.7·24-s − 180.·26-s − 109.·27-s − 60.9·28-s + 94.5·29-s − 178.·31-s + 32·32-s + 71.7·33-s − 167.·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.14·3-s + 0.5·4-s + 0.812·6-s − 0.823·7-s + 0.353·8-s + 0.319·9-s + 0.329·11-s + 0.574·12-s − 1.92·13-s − 0.582·14-s + 0.250·16-s − 1.19·17-s + 0.225·18-s + 0.903·19-s − 0.945·21-s + 0.233·22-s − 0.208·23-s + 0.406·24-s − 1.36·26-s − 0.781·27-s − 0.411·28-s + 0.605·29-s − 1.03·31-s + 0.176·32-s + 0.378·33-s − 0.847·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 - 5.96T + 27T^{2} \) |
| 7 | \( 1 + 15.2T + 343T^{2} \) |
| 11 | \( 1 - 12.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 90.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 83.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 74.8T + 6.85e3T^{2} \) |
| 29 | \( 1 - 94.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 178.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 296.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 193.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 15.4T + 7.95e4T^{2} \) |
| 47 | \( 1 + 128.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 342.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 725.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 784.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 782.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 141.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 404.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 721.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 985.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 463.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.68e3T + 9.12e5T^{2} \) |
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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
−9.216744744416754607486336350974, −8.113872677859011630557339241931, −7.25760443123851139237460294481, −6.66422259294030883552663782433, −5.46994235453718755493871845649, −4.54163034909032702287598342483, −3.51216909322230850278039185208, −2.77628966699808700922923028034, −1.97781963216675973444988455804, 0,
1.97781963216675973444988455804, 2.77628966699808700922923028034, 3.51216909322230850278039185208, 4.54163034909032702287598342483, 5.46994235453718755493871845649, 6.66422259294030883552663782433, 7.25760443123851139237460294481, 8.113872677859011630557339241931, 9.216744744416754607486336350974
