| L(s) = 1 | − 1.66·2-s + 2.43·3-s + 0.779·4-s + 5-s − 4.06·6-s + 7-s + 2.03·8-s + 2.94·9-s − 1.66·10-s + 3.17·11-s + 1.90·12-s − 2.08·13-s − 1.66·14-s + 2.43·15-s − 4.95·16-s − 2.91·17-s − 4.91·18-s − 5.66·19-s + 0.779·20-s + 2.43·21-s − 5.28·22-s − 1.71·23-s + 4.96·24-s + 25-s + 3.48·26-s − 0.127·27-s + 0.779·28-s + ⋯ |
| L(s) = 1 | − 1.17·2-s + 1.40·3-s + 0.389·4-s + 0.447·5-s − 1.65·6-s + 0.377·7-s + 0.719·8-s + 0.982·9-s − 0.527·10-s + 0.956·11-s + 0.548·12-s − 0.579·13-s − 0.445·14-s + 0.629·15-s − 1.23·16-s − 0.707·17-s − 1.15·18-s − 1.30·19-s + 0.174·20-s + 0.532·21-s − 1.12·22-s − 0.357·23-s + 1.01·24-s + 0.200·25-s + 0.683·26-s − 0.0246·27-s + 0.147·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 + 1.66T + 2T^{2} \) |
| 3 | \( 1 - 2.43T + 3T^{2} \) |
| 11 | \( 1 - 3.17T + 11T^{2} \) |
| 13 | \( 1 + 2.08T + 13T^{2} \) |
| 17 | \( 1 + 2.91T + 17T^{2} \) |
| 19 | \( 1 + 5.66T + 19T^{2} \) |
| 23 | \( 1 + 1.71T + 23T^{2} \) |
| 29 | \( 1 + 5.05T + 29T^{2} \) |
| 31 | \( 1 + 3.30T + 31T^{2} \) |
| 37 | \( 1 + 3.21T + 37T^{2} \) |
| 41 | \( 1 - 3.49T + 41T^{2} \) |
| 43 | \( 1 - 2.26T + 43T^{2} \) |
| 47 | \( 1 + 1.59T + 47T^{2} \) |
| 53 | \( 1 + 0.825T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 - 8.83T + 61T^{2} \) |
| 67 | \( 1 - 0.605T + 67T^{2} \) |
| 71 | \( 1 + 6.38T + 71T^{2} \) |
| 73 | \( 1 + 8.89T + 73T^{2} \) |
| 79 | \( 1 - 8.19T + 79T^{2} \) |
| 83 | \( 1 + 4.56T + 83T^{2} \) |
| 89 | \( 1 + 6.31T + 89T^{2} \) |
| 97 | \( 1 - 9.41T + 97T^{2} \) |
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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
−7.78502373777500333138913315554, −7.09663474547860676756107059452, −6.47407890897391751810052294245, −5.40986324390668260749822062607, −4.31810729247615778434316990049, −3.97668543065335586923768771993, −2.75538450907634350620042957482, −1.98758885321494966718864069814, −1.52682136184330370549986226616, 0,
1.52682136184330370549986226616, 1.98758885321494966718864069814, 2.75538450907634350620042957482, 3.97668543065335586923768771993, 4.31810729247615778434316990049, 5.40986324390668260749822062607, 6.47407890897391751810052294245, 7.09663474547860676756107059452, 7.78502373777500333138913315554
