| L(s) = 1 | + 1.16·2-s − 0.371·3-s − 0.651·4-s + 0.0627·5-s − 0.431·6-s + 2.77·7-s − 3.07·8-s − 2.86·9-s + 0.0728·10-s + 11-s + 0.242·12-s − 3.13·13-s + 3.22·14-s − 0.0233·15-s − 2.27·16-s − 17-s − 3.32·18-s − 7.56·19-s − 0.0409·20-s − 1.03·21-s + 1.16·22-s + 0.329·23-s + 1.14·24-s − 4.99·25-s − 3.63·26-s + 2.18·27-s − 1.81·28-s + ⋯ |
| L(s) = 1 | + 0.821·2-s − 0.214·3-s − 0.325·4-s + 0.0280·5-s − 0.176·6-s + 1.04·7-s − 1.08·8-s − 0.953·9-s + 0.0230·10-s + 0.301·11-s + 0.0699·12-s − 0.868·13-s + 0.862·14-s − 0.00602·15-s − 0.567·16-s − 0.242·17-s − 0.783·18-s − 1.73·19-s − 0.00914·20-s − 0.225·21-s + 0.247·22-s + 0.0687·23-s + 0.233·24-s − 0.999·25-s − 0.712·26-s + 0.419·27-s − 0.342·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.683326182\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.683326182\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 - 1.16T + 2T^{2} \) |
| 3 | \( 1 + 0.371T + 3T^{2} \) |
| 5 | \( 1 - 0.0627T + 5T^{2} \) |
| 7 | \( 1 - 2.77T + 7T^{2} \) |
| 13 | \( 1 + 3.13T + 13T^{2} \) |
| 19 | \( 1 + 7.56T + 19T^{2} \) |
| 23 | \( 1 - 0.329T + 23T^{2} \) |
| 29 | \( 1 - 0.615T + 29T^{2} \) |
| 31 | \( 1 - 7.88T + 31T^{2} \) |
| 37 | \( 1 + 0.492T + 37T^{2} \) |
| 41 | \( 1 + 3.81T + 41T^{2} \) |
| 47 | \( 1 - 7.08T + 47T^{2} \) |
| 53 | \( 1 - 5.59T + 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 - 2.02T + 61T^{2} \) |
| 67 | \( 1 - 1.39T + 67T^{2} \) |
| 71 | \( 1 + 7.32T + 71T^{2} \) |
| 73 | \( 1 - 7.39T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 7.52T + 83T^{2} \) |
| 89 | \( 1 + 1.79T + 89T^{2} \) |
| 97 | \( 1 - 1.32T + 97T^{2} \) |
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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
−8.017525968595496774027521847906, −6.92737469034356737836756589564, −6.25044705675035294047632233735, −5.60490380231895334409508848967, −4.95330186256638793025053224458, −4.41587464302859869789382401672, −3.76887412027933312201510430748, −2.66816400384322568559163251995, −2.06118027715420159509699217634, −0.54981126829773471743828218407,
0.54981126829773471743828218407, 2.06118027715420159509699217634, 2.66816400384322568559163251995, 3.76887412027933312201510430748, 4.41587464302859869789382401672, 4.95330186256638793025053224458, 5.60490380231895334409508848967, 6.25044705675035294047632233735, 6.92737469034356737836756589564, 8.017525968595496774027521847906
