| L(s) = 1 | + 2-s − 2.88·3-s + 4-s + 5-s − 2.88·6-s − 2.43·7-s + 8-s + 5.32·9-s + 10-s − 11-s − 2.88·12-s − 1.17·13-s − 2.43·14-s − 2.88·15-s + 16-s − 6.19·17-s + 5.32·18-s + 6.20·19-s + 20-s + 7.03·21-s − 22-s + 3.20·23-s − 2.88·24-s + 25-s − 1.17·26-s − 6.69·27-s − 2.43·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.66·3-s + 0.5·4-s + 0.447·5-s − 1.17·6-s − 0.921·7-s + 0.353·8-s + 1.77·9-s + 0.316·10-s − 0.301·11-s − 0.832·12-s − 0.326·13-s − 0.651·14-s − 0.744·15-s + 0.250·16-s − 1.50·17-s + 1.25·18-s + 1.42·19-s + 0.223·20-s + 1.53·21-s − 0.213·22-s + 0.668·23-s − 0.588·24-s + 0.200·25-s − 0.230·26-s − 1.28·27-s − 0.460·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 + 2.88T + 3T^{2} \) |
| 7 | \( 1 + 2.43T + 7T^{2} \) |
| 13 | \( 1 + 1.17T + 13T^{2} \) |
| 17 | \( 1 + 6.19T + 17T^{2} \) |
| 19 | \( 1 - 6.20T + 19T^{2} \) |
| 23 | \( 1 - 3.20T + 23T^{2} \) |
| 29 | \( 1 + 0.624T + 29T^{2} \) |
| 31 | \( 1 - 5.87T + 31T^{2} \) |
| 37 | \( 1 - 1.00T + 37T^{2} \) |
| 41 | \( 1 - 0.823T + 41T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 + 5.91T + 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 - 9.88T + 71T^{2} \) |
| 73 | \( 1 + 1.73T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 2.14T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−7.44507826893978226429225794379, −6.91748807870275126011207782105, −6.23886128734988025477678831529, −5.85943379148380675994214807620, −4.90267852061984151339419887687, −4.65064135012599589194183051629, −3.40952431233701584037872339800, −2.52266329568597306040936601062, −1.22946611126645627911295550013, 0,
1.22946611126645627911295550013, 2.52266329568597306040936601062, 3.40952431233701584037872339800, 4.65064135012599589194183051629, 4.90267852061984151339419887687, 5.85943379148380675994214807620, 6.23886128734988025477678831529, 6.91748807870275126011207782105, 7.44507826893978226429225794379
