| L(s) = 1 | + 0.700·2-s − 1.90·3-s − 1.50·4-s − 1.33·6-s − 7-s − 2.45·8-s + 0.638·9-s + 1.45·11-s + 2.87·12-s + 13-s − 0.700·14-s + 1.29·16-s + 6.90·17-s + 0.447·18-s − 2.77·19-s + 1.90·21-s + 1.01·22-s + 5.28·23-s + 4.68·24-s + 0.700·26-s + 4.50·27-s + 1.50·28-s + 0.605·29-s − 4.55·31-s + 5.82·32-s − 2.77·33-s + 4.83·34-s + ⋯ |
| L(s) = 1 | + 0.495·2-s − 1.10·3-s − 0.754·4-s − 0.545·6-s − 0.377·7-s − 0.868·8-s + 0.212·9-s + 0.438·11-s + 0.831·12-s + 0.277·13-s − 0.187·14-s + 0.324·16-s + 1.67·17-s + 0.105·18-s − 0.635·19-s + 0.416·21-s + 0.217·22-s + 1.10·23-s + 0.956·24-s + 0.137·26-s + 0.866·27-s + 0.285·28-s + 0.112·29-s − 0.817·31-s + 1.02·32-s − 0.483·33-s + 0.829·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 0.700T + 2T^{2} \) |
| 3 | \( 1 + 1.90T + 3T^{2} \) |
| 11 | \( 1 - 1.45T + 11T^{2} \) |
| 17 | \( 1 - 6.90T + 17T^{2} \) |
| 19 | \( 1 + 2.77T + 19T^{2} \) |
| 23 | \( 1 - 5.28T + 23T^{2} \) |
| 29 | \( 1 - 0.605T + 29T^{2} \) |
| 31 | \( 1 + 4.55T + 31T^{2} \) |
| 37 | \( 1 + 6.64T + 37T^{2} \) |
| 41 | \( 1 + 8.10T + 41T^{2} \) |
| 43 | \( 1 + 2.01T + 43T^{2} \) |
| 47 | \( 1 - 0.402T + 47T^{2} \) |
| 53 | \( 1 + 6.04T + 53T^{2} \) |
| 59 | \( 1 - 0.173T + 59T^{2} \) |
| 61 | \( 1 + 12.8T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 - 16.3T + 71T^{2} \) |
| 73 | \( 1 - 1.79T + 73T^{2} \) |
| 79 | \( 1 + 3.07T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 + 0.0343T + 89T^{2} \) |
| 97 | \( 1 - 2.49T + 97T^{2} \) |
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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
−8.767580580990432215405277972798, −7.84985803991526902583535545551, −6.73856946687567657935837513087, −6.13545135852387879770076929865, −5.33588588827434530238236239950, −4.90474501402797479924355663739, −3.75010091118653692364447715788, −3.11096523278524776425243204055, −1.25294580133689516146825860771, 0,
1.25294580133689516146825860771, 3.11096523278524776425243204055, 3.75010091118653692364447715788, 4.90474501402797479924355663739, 5.33588588827434530238236239950, 6.13545135852387879770076929865, 6.73856946687567657935837513087, 7.84985803991526902583535545551, 8.767580580990432215405277972798
