| L(s) = 1 | + 2.42·2-s − 3-s + 3.86·4-s − 2.93·5-s − 2.42·6-s + 4.51·8-s + 9-s − 7.11·10-s − 1.07·11-s − 3.86·12-s + 2.25·13-s + 2.93·15-s + 3.20·16-s + 1.82·17-s + 2.42·18-s − 5.97·19-s − 11.3·20-s − 2.60·22-s − 23-s − 4.51·24-s + 3.63·25-s + 5.47·26-s − 27-s + 2.11·29-s + 7.11·30-s − 4.79·31-s − 1.27·32-s + ⋯ |
| L(s) = 1 | + 1.71·2-s − 0.577·3-s + 1.93·4-s − 1.31·5-s − 0.988·6-s + 1.59·8-s + 0.333·9-s − 2.25·10-s − 0.324·11-s − 1.11·12-s + 0.626·13-s + 0.758·15-s + 0.800·16-s + 0.442·17-s + 0.570·18-s − 1.37·19-s − 2.53·20-s − 0.555·22-s − 0.208·23-s − 0.921·24-s + 0.727·25-s + 1.07·26-s − 0.192·27-s + 0.393·29-s + 1.29·30-s − 0.861·31-s − 0.225·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 2.42T + 2T^{2} \) |
| 5 | \( 1 + 2.93T + 5T^{2} \) |
| 11 | \( 1 + 1.07T + 11T^{2} \) |
| 13 | \( 1 - 2.25T + 13T^{2} \) |
| 17 | \( 1 - 1.82T + 17T^{2} \) |
| 19 | \( 1 + 5.97T + 19T^{2} \) |
| 29 | \( 1 - 2.11T + 29T^{2} \) |
| 31 | \( 1 + 4.79T + 31T^{2} \) |
| 37 | \( 1 - 0.784T + 37T^{2} \) |
| 41 | \( 1 + 0.651T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 - 5.13T + 53T^{2} \) |
| 59 | \( 1 - 4.95T + 59T^{2} \) |
| 61 | \( 1 + 6.70T + 61T^{2} \) |
| 67 | \( 1 + 4.90T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 6.14T + 73T^{2} \) |
| 79 | \( 1 + 4.65T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 - 7.98T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−8.021017236210675547068164861432, −7.18714349281996535044382790825, −6.53379820712832685122765849294, −5.85231761906690338026928937843, −5.01456661551282903869792686553, −4.36477888044020493304304441499, −3.73916820012787259615740284230, −3.04587574386962322048715472501, −1.75687181444058684189420896321, 0,
1.75687181444058684189420896321, 3.04587574386962322048715472501, 3.73916820012787259615740284230, 4.36477888044020493304304441499, 5.01456661551282903869792686553, 5.85231761906690338026928937843, 6.53379820712832685122765849294, 7.18714349281996535044382790825, 8.021017236210675547068164861432
