| L(s) = 1 | + 2-s − 1.56·3-s + 4-s − 5-s − 1.56·6-s − 1.49·7-s + 8-s − 0.540·9-s − 10-s − 1.02·11-s − 1.56·12-s + 13-s − 1.49·14-s + 1.56·15-s + 16-s + 1.43·17-s − 0.540·18-s + 2.94·19-s − 20-s + 2.33·21-s − 1.02·22-s + 1.14·23-s − 1.56·24-s + 25-s + 26-s + 5.55·27-s − 1.49·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.905·3-s + 0.5·4-s − 0.447·5-s − 0.640·6-s − 0.563·7-s + 0.353·8-s − 0.180·9-s − 0.316·10-s − 0.309·11-s − 0.452·12-s + 0.277·13-s − 0.398·14-s + 0.404·15-s + 0.250·16-s + 0.347·17-s − 0.127·18-s + 0.675·19-s − 0.223·20-s + 0.510·21-s − 0.218·22-s + 0.237·23-s − 0.320·24-s + 0.200·25-s + 0.196·26-s + 1.06·27-s − 0.281·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 7 | \( 1 + 1.49T + 7T^{2} \) |
| 11 | \( 1 + 1.02T + 11T^{2} \) |
| 17 | \( 1 - 1.43T + 17T^{2} \) |
| 19 | \( 1 - 2.94T + 19T^{2} \) |
| 23 | \( 1 - 1.14T + 23T^{2} \) |
| 29 | \( 1 - 6.02T + 29T^{2} \) |
| 37 | \( 1 + 6.22T + 37T^{2} \) |
| 41 | \( 1 - 3.45T + 41T^{2} \) |
| 43 | \( 1 - 4.32T + 43T^{2} \) |
| 47 | \( 1 + 4.37T + 47T^{2} \) |
| 53 | \( 1 + 9.04T + 53T^{2} \) |
| 59 | \( 1 + 8.83T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 9.32T + 67T^{2} \) |
| 71 | \( 1 + 15.9T + 71T^{2} \) |
| 73 | \( 1 + 4.75T + 73T^{2} \) |
| 79 | \( 1 + 5.97T + 79T^{2} \) |
| 83 | \( 1 - 0.949T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−7.898242782008704109030244986810, −7.15854777946553951581904272649, −6.37570296935615149417514420209, −5.86999206013247433929774848894, −5.07357707870403998563344845520, −4.46978052464605268275228356593, −3.34345103794946846890114531416, −2.85624532151780667063459694251, −1.30999270594204474368782989440, 0,
1.30999270594204474368782989440, 2.85624532151780667063459694251, 3.34345103794946846890114531416, 4.46978052464605268275228356593, 5.07357707870403998563344845520, 5.86999206013247433929774848894, 6.37570296935615149417514420209, 7.15854777946553951581904272649, 7.898242782008704109030244986810
