| L(s) = 1 | + 2.70·2-s + 6.22·3-s − 0.677·4-s + 5·5-s + 16.8·6-s − 21.7·7-s − 23.4·8-s + 11.7·9-s + 13.5·10-s + 11·11-s − 4.21·12-s + 67.9·13-s − 58.8·14-s + 31.1·15-s − 58.1·16-s + 62.9·17-s + 31.7·18-s − 19·19-s − 3.38·20-s − 135.·21-s + 29.7·22-s + 183.·23-s − 146.·24-s + 25·25-s + 183.·26-s − 95.0·27-s + 14.7·28-s + ⋯ |
| L(s) = 1 | + 0.956·2-s + 1.19·3-s − 0.0847·4-s + 0.447·5-s + 1.14·6-s − 1.17·7-s − 1.03·8-s + 0.434·9-s + 0.427·10-s + 0.301·11-s − 0.101·12-s + 1.44·13-s − 1.12·14-s + 0.535·15-s − 0.908·16-s + 0.898·17-s + 0.415·18-s − 0.229·19-s − 0.0378·20-s − 1.40·21-s + 0.288·22-s + 1.66·23-s − 1.24·24-s + 0.200·25-s + 1.38·26-s − 0.677·27-s + 0.0995·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.842528197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.842528197\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 - 2.70T + 8T^{2} \) |
| 3 | \( 1 - 6.22T + 27T^{2} \) |
| 7 | \( 1 + 21.7T + 343T^{2} \) |
| 13 | \( 1 - 67.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 62.9T + 4.91e3T^{2} \) |
| 23 | \( 1 - 183.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 136.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 283.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 30.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 64.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 0.464T + 7.95e4T^{2} \) |
| 47 | \( 1 + 4.97T + 1.03e5T^{2} \) |
| 53 | \( 1 + 89.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + 152.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 224.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 2.06T + 3.00e5T^{2} \) |
| 71 | \( 1 - 487.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.02e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 375.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 392.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 460.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 244.T + 9.12e5T^{2} \) |
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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
−9.373599452955073691113942841967, −8.816936003428632438143888024295, −8.107120759034417713278420896462, −6.67083728451580638939894085224, −6.17742071652627250804052598374, −5.14543055975326871113374947819, −3.94296162937562870633662311347, −3.25182149105166836308892563976, −2.70432861430571808410137360012, −0.991109396966758846682091711777,
0.991109396966758846682091711777, 2.70432861430571808410137360012, 3.25182149105166836308892563976, 3.94296162937562870633662311347, 5.14543055975326871113374947819, 6.17742071652627250804052598374, 6.67083728451580638939894085224, 8.107120759034417713278420896462, 8.816936003428632438143888024295, 9.373599452955073691113942841967
