| L(s) = 1 | + 0.465·2-s − 3-s − 1.78·4-s − 2.88·5-s − 0.465·6-s + 0.971·7-s − 1.75·8-s + 9-s − 1.34·10-s − 11-s + 1.78·12-s + 2.82·13-s + 0.452·14-s + 2.88·15-s + 2.74·16-s + 2.54·17-s + 0.465·18-s − 6.80·19-s + 5.14·20-s − 0.971·21-s − 0.465·22-s + 0.244·23-s + 1.75·24-s + 3.32·25-s + 1.31·26-s − 27-s − 1.73·28-s + ⋯ |
| L(s) = 1 | + 0.328·2-s − 0.577·3-s − 0.891·4-s − 1.29·5-s − 0.189·6-s + 0.367·7-s − 0.622·8-s + 0.333·9-s − 0.424·10-s − 0.301·11-s + 0.514·12-s + 0.784·13-s + 0.120·14-s + 0.745·15-s + 0.687·16-s + 0.616·17-s + 0.109·18-s − 1.56·19-s + 1.15·20-s − 0.212·21-s − 0.0991·22-s + 0.0509·23-s + 0.359·24-s + 0.665·25-s + 0.258·26-s − 0.192·27-s − 0.327·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 97 | \( 1 + T \) |
| good | 2 | \( 1 - 0.465T + 2T^{2} \) |
| 5 | \( 1 + 2.88T + 5T^{2} \) |
| 7 | \( 1 - 0.971T + 7T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 - 2.54T + 17T^{2} \) |
| 19 | \( 1 + 6.80T + 19T^{2} \) |
| 23 | \( 1 - 0.244T + 23T^{2} \) |
| 29 | \( 1 - 7.99T + 29T^{2} \) |
| 31 | \( 1 - 1.83T + 31T^{2} \) |
| 37 | \( 1 - 7.78T + 37T^{2} \) |
| 41 | \( 1 + 3.54T + 41T^{2} \) |
| 43 | \( 1 - 0.693T + 43T^{2} \) |
| 47 | \( 1 + 3.49T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 - 1.86T + 59T^{2} \) |
| 61 | \( 1 - 1.89T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 - 16.4T + 71T^{2} \) |
| 73 | \( 1 + 5.70T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 + 0.485T + 89T^{2} \) |
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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
−8.355857441657638105248374353637, −7.72030532502944673359225533868, −6.67664583824446901405471373119, −5.93949453320798834940262175494, −5.04742165108425864641567111269, −4.31974080214466426798979351364, −3.90548677436630130747384851934, −2.83972225784073975280884159321, −1.11213121918358566759610583060, 0,
1.11213121918358566759610583060, 2.83972225784073975280884159321, 3.90548677436630130747384851934, 4.31974080214466426798979351364, 5.04742165108425864641567111269, 5.93949453320798834940262175494, 6.67664583824446901405471373119, 7.72030532502944673359225533868, 8.355857441657638105248374353637
