| L(s) = 1 | + 1.51·2-s + 1.55·3-s + 0.291·4-s + 2.35·6-s − 7-s − 2.58·8-s − 0.577·9-s − 1.70·11-s + 0.453·12-s + 13-s − 1.51·14-s − 4.49·16-s − 4.39·17-s − 0.873·18-s − 5.08·19-s − 1.55·21-s − 2.58·22-s − 5.25·23-s − 4.02·24-s + 1.51·26-s − 5.56·27-s − 0.291·28-s + 0.226·29-s − 1.70·31-s − 1.63·32-s − 2.65·33-s − 6.65·34-s + ⋯ |
| L(s) = 1 | + 1.07·2-s + 0.898·3-s + 0.145·4-s + 0.961·6-s − 0.377·7-s − 0.914·8-s − 0.192·9-s − 0.514·11-s + 0.130·12-s + 0.277·13-s − 0.404·14-s − 1.12·16-s − 1.06·17-s − 0.205·18-s − 1.16·19-s − 0.339·21-s − 0.550·22-s − 1.09·23-s − 0.821·24-s + 0.296·26-s − 1.07·27-s − 0.0550·28-s + 0.0420·29-s − 0.306·31-s − 0.289·32-s − 0.462·33-s − 1.14·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 1.51T + 2T^{2} \) |
| 3 | \( 1 - 1.55T + 3T^{2} \) |
| 11 | \( 1 + 1.70T + 11T^{2} \) |
| 17 | \( 1 + 4.39T + 17T^{2} \) |
| 19 | \( 1 + 5.08T + 19T^{2} \) |
| 23 | \( 1 + 5.25T + 23T^{2} \) |
| 29 | \( 1 - 0.226T + 29T^{2} \) |
| 31 | \( 1 + 1.70T + 31T^{2} \) |
| 37 | \( 1 - 5.41T + 37T^{2} \) |
| 41 | \( 1 + 0.800T + 41T^{2} \) |
| 43 | \( 1 - 1.43T + 43T^{2} \) |
| 47 | \( 1 - 9.52T + 47T^{2} \) |
| 53 | \( 1 + 0.943T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 - 5.89T + 61T^{2} \) |
| 67 | \( 1 + 9.53T + 67T^{2} \) |
| 71 | \( 1 + 1.82T + 71T^{2} \) |
| 73 | \( 1 + 6.40T + 73T^{2} \) |
| 79 | \( 1 - 17.3T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 + 17.9T + 89T^{2} \) |
| 97 | \( 1 + 6.64T + 97T^{2} \) |
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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
−8.647370275389387680321361867766, −8.027392772421794366521371551262, −6.91724384639134429940020406036, −6.09167341502014503368242833646, −5.47134471933677466044711826376, −4.27957852245877597477580014738, −3.89091866453475024972536871481, −2.78863101453598738858342471141, −2.23749905434210055217015271629, 0,
2.23749905434210055217015271629, 2.78863101453598738858342471141, 3.89091866453475024972536871481, 4.27957852245877597477580014738, 5.47134471933677466044711826376, 6.09167341502014503368242833646, 6.91724384639134429940020406036, 8.027392772421794366521371551262, 8.647370275389387680321361867766
