| L(s) = 1 | − 3-s + 0.900·5-s + 3.90·7-s + 9-s − 5.39·11-s − 2.18·13-s − 0.900·15-s − 2.64·17-s + 3.37·19-s − 3.90·21-s − 23-s − 4.18·25-s − 27-s + 29-s + 5.08·31-s + 5.39·33-s + 3.51·35-s − 3.79·37-s + 2.18·39-s + 8.82·41-s + 0.613·43-s + 0.900·45-s + 1.02·47-s + 8.24·49-s + 2.64·51-s + 5.94·53-s − 4.86·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.402·5-s + 1.47·7-s + 0.333·9-s − 1.62·11-s − 0.606·13-s − 0.232·15-s − 0.641·17-s + 0.773·19-s − 0.852·21-s − 0.208·23-s − 0.837·25-s − 0.192·27-s + 0.185·29-s + 0.913·31-s + 0.939·33-s + 0.594·35-s − 0.623·37-s + 0.349·39-s + 1.37·41-s + 0.0935·43-s + 0.134·45-s + 0.149·47-s + 1.17·49-s + 0.370·51-s + 0.816·53-s − 0.655·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 0.900T + 5T^{2} \) |
| 7 | \( 1 - 3.90T + 7T^{2} \) |
| 11 | \( 1 + 5.39T + 11T^{2} \) |
| 13 | \( 1 + 2.18T + 13T^{2} \) |
| 17 | \( 1 + 2.64T + 17T^{2} \) |
| 19 | \( 1 - 3.37T + 19T^{2} \) |
| 31 | \( 1 - 5.08T + 31T^{2} \) |
| 37 | \( 1 + 3.79T + 37T^{2} \) |
| 41 | \( 1 - 8.82T + 41T^{2} \) |
| 43 | \( 1 - 0.613T + 43T^{2} \) |
| 47 | \( 1 - 1.02T + 47T^{2} \) |
| 53 | \( 1 - 5.94T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 + 3.09T + 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 + 4.69T + 71T^{2} \) |
| 73 | \( 1 - 9.03T + 73T^{2} \) |
| 79 | \( 1 + 3.30T + 79T^{2} \) |
| 83 | \( 1 + 1.01T + 83T^{2} \) |
| 89 | \( 1 + 17.4T + 89T^{2} \) |
| 97 | \( 1 - 4.91T + 97T^{2} \) |
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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
−7.68728753430991665362881938926, −6.87309612414701422486535133353, −5.81057178030045534614511799534, −5.44628066567346403393969939634, −4.74430304205988557608171391378, −4.26746481367675191393026223490, −2.85326570971119426892813440256, −2.20027547337030882009090717352, −1.29771476150099003287640726054, 0,
1.29771476150099003287640726054, 2.20027547337030882009090717352, 2.85326570971119426892813440256, 4.26746481367675191393026223490, 4.74430304205988557608171391378, 5.44628066567346403393969939634, 5.81057178030045534614511799534, 6.87309612414701422486535133353, 7.68728753430991665362881938926
