| L(s) = 1 | − 2.90·2-s + 6.23·3-s + 0.417·4-s + 15.8·5-s − 18.1·6-s − 9.46·7-s + 21.9·8-s + 11.9·9-s − 45.8·10-s − 11·11-s + 2.60·12-s + 27.4·14-s + 98.5·15-s − 67.1·16-s + 103.·17-s − 34.5·18-s − 148.·19-s + 6.58·20-s − 59.0·21-s + 31.9·22-s − 5.09·23-s + 137.·24-s + 124.·25-s − 94.0·27-s − 3.94·28-s − 285.·29-s − 285.·30-s + ⋯ |
| L(s) = 1 | − 1.02·2-s + 1.20·3-s + 0.0521·4-s + 1.41·5-s − 1.23·6-s − 0.511·7-s + 0.972·8-s + 0.441·9-s − 1.44·10-s − 0.301·11-s + 0.0625·12-s + 0.524·14-s + 1.69·15-s − 1.04·16-s + 1.47·17-s − 0.452·18-s − 1.79·19-s + 0.0736·20-s − 0.613·21-s + 0.309·22-s − 0.0461·23-s + 1.16·24-s + 0.997·25-s − 0.670·27-s − 0.0266·28-s − 1.83·29-s − 1.74·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.90T + 8T^{2} \) |
| 3 | \( 1 - 6.23T + 27T^{2} \) |
| 5 | \( 1 - 15.8T + 125T^{2} \) |
| 7 | \( 1 + 9.46T + 343T^{2} \) |
| 17 | \( 1 - 103.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 148.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 5.09T + 1.21e4T^{2} \) |
| 29 | \( 1 + 285.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 274.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 303.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 47.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 141.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 575.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 342.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 69.2T + 2.05e5T^{2} \) |
| 61 | \( 1 + 360.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 190.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 311.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 317.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.14e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.19e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 117.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 597.T + 9.12e5T^{2} \) |
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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
−8.671018728120756124615296214931, −8.013455380105521209908925197441, −7.21875009119952711484574703645, −6.18208951067782695965009986917, −5.40597062742100548322268762034, −4.17997039852649519501446631247, −3.09839891901438315798335551031, −2.17194883573689467903049696168, −1.47527454503862514670030563608, 0,
1.47527454503862514670030563608, 2.17194883573689467903049696168, 3.09839891901438315798335551031, 4.17997039852649519501446631247, 5.40597062742100548322268762034, 6.18208951067782695965009986917, 7.21875009119952711484574703645, 8.013455380105521209908925197441, 8.671018728120756124615296214931
