| L(s) = 1 | + 2.47·2-s − 1.89·4-s + 11.8·5-s + 15.4·7-s − 24.4·8-s + 29.2·10-s − 8.94·11-s − 62.7·13-s + 38.1·14-s − 45.2·16-s + 66.9·17-s − 20.0·19-s − 22.4·20-s − 22.0·22-s − 21.6·23-s + 14.9·25-s − 155.·26-s − 29.3·28-s + 5.86·29-s − 327.·31-s + 83.9·32-s + 165.·34-s + 182.·35-s − 44.5·37-s − 49.4·38-s − 289.·40-s − 13.6·41-s + ⋯ |
| L(s) = 1 | + 0.873·2-s − 0.237·4-s + 1.05·5-s + 0.833·7-s − 1.08·8-s + 0.923·10-s − 0.245·11-s − 1.33·13-s + 0.727·14-s − 0.706·16-s + 0.955·17-s − 0.241·19-s − 0.251·20-s − 0.214·22-s − 0.196·23-s + 0.119·25-s − 1.16·26-s − 0.197·28-s + 0.0375·29-s − 1.89·31-s + 0.463·32-s + 0.834·34-s + 0.881·35-s − 0.198·37-s − 0.211·38-s − 1.14·40-s − 0.0518·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1503 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 167 | \( 1 + 167T \) |
| good | 2 | \( 1 - 2.47T + 8T^{2} \) |
| 5 | \( 1 - 11.8T + 125T^{2} \) |
| 7 | \( 1 - 15.4T + 343T^{2} \) |
| 11 | \( 1 + 8.94T + 1.33e3T^{2} \) |
| 13 | \( 1 + 62.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 66.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 20.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 21.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 5.86T + 2.43e4T^{2} \) |
| 31 | \( 1 + 327.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 44.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 13.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 106.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 264.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 280.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 701.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 4.83T + 2.26e5T^{2} \) |
| 67 | \( 1 - 371.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 402.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 231.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 693.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.13e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 397.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 349.T + 9.12e5T^{2} \) |
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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
−8.769413668840211027255688587043, −7.88809053866230459502331083165, −6.97369813661645399062184415436, −5.85437877701068266212441634044, −5.31349041623888966640211533710, −4.74285441926086665466980326427, −3.64463815919644494244620130308, −2.55444524063963309674713405781, −1.63483224222363097802948307945, 0,
1.63483224222363097802948307945, 2.55444524063963309674713405781, 3.64463815919644494244620130308, 4.74285441926086665466980326427, 5.31349041623888966640211533710, 5.85437877701068266212441634044, 6.97369813661645399062184415436, 7.88809053866230459502331083165, 8.769413668840211027255688587043
