L(s) = 1 | − 2-s + 2.42·3-s + 4-s − 0.260·5-s − 2.42·6-s − 8-s + 2.88·9-s + 0.260·10-s + 2.14·11-s + 2.42·12-s − 6.34·13-s − 0.632·15-s + 16-s − 2.22·17-s − 2.88·18-s − 8.43·19-s − 0.260·20-s − 2.14·22-s + 4.38·23-s − 2.42·24-s − 4.93·25-s + 6.34·26-s − 0.274·27-s − 29-s + 0.632·30-s + 3.04·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.40·3-s + 0.5·4-s − 0.116·5-s − 0.990·6-s − 0.353·8-s + 0.962·9-s + 0.0824·10-s + 0.647·11-s + 0.700·12-s − 1.75·13-s − 0.163·15-s + 0.250·16-s − 0.540·17-s − 0.680·18-s − 1.93·19-s − 0.0583·20-s − 0.457·22-s + 0.914·23-s − 0.495·24-s − 0.986·25-s + 1.24·26-s − 0.0528·27-s − 0.185·29-s + 0.115·30-s + 0.546·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 29 | \( 1 + T \) |
good | 3 | \( 1 - 2.42T + 3T^{2} \) |
| 5 | \( 1 + 0.260T + 5T^{2} \) |
| 11 | \( 1 - 2.14T + 11T^{2} \) |
| 13 | \( 1 + 6.34T + 13T^{2} \) |
| 17 | \( 1 + 2.22T + 17T^{2} \) |
| 19 | \( 1 + 8.43T + 19T^{2} \) |
| 23 | \( 1 - 4.38T + 23T^{2} \) |
| 31 | \( 1 - 3.04T + 31T^{2} \) |
| 37 | \( 1 + 7.24T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 - 0.195T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 + 0.390T + 53T^{2} \) |
| 59 | \( 1 - 14.7T + 59T^{2} \) |
| 61 | \( 1 - 9.83T + 61T^{2} \) |
| 67 | \( 1 - 9.53T + 67T^{2} \) |
| 71 | \( 1 - 2.18T + 71T^{2} \) |
| 73 | \( 1 + 4.18T + 73T^{2} \) |
| 79 | \( 1 + 4.48T + 79T^{2} \) |
| 83 | \( 1 + 1.91T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 - 0.283T + 97T^{2} \) |
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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
−8.553956315096583971101724862507, −7.912616983592737138072445810780, −6.98092775270042140545544736576, −6.63914454975993484410180241748, −5.20948619483113184034008908366, −4.24888724408091617151288131672, −3.38019591522368905582936580733, −2.37366976238035824048389540655, −1.86890974327642007631453148872, 0,
1.86890974327642007631453148872, 2.37366976238035824048389540655, 3.38019591522368905582936580733, 4.24888724408091617151288131672, 5.20948619483113184034008908366, 6.63914454975993484410180241748, 6.98092775270042140545544736576, 7.912616983592737138072445810780, 8.553956315096583971101724862507