| L(s) = 1 | − 2.30·2-s − 0.930·3-s + 3.31·4-s + 2.14·6-s − 7-s − 3.02·8-s − 2.13·9-s + 0.709·11-s − 3.08·12-s − 3.91·13-s + 2.30·14-s + 0.354·16-s − 4.79·17-s + 4.91·18-s + 2.72·19-s + 0.930·21-s − 1.63·22-s − 23-s + 2.81·24-s + 9.03·26-s + 4.77·27-s − 3.31·28-s + 9.19·29-s + 4.59·31-s + 5.24·32-s − 0.660·33-s + 11.0·34-s + ⋯ |
| L(s) = 1 | − 1.63·2-s − 0.537·3-s + 1.65·4-s + 0.875·6-s − 0.377·7-s − 1.07·8-s − 0.711·9-s + 0.213·11-s − 0.890·12-s − 1.08·13-s + 0.616·14-s + 0.0885·16-s − 1.16·17-s + 1.15·18-s + 0.625·19-s + 0.203·21-s − 0.348·22-s − 0.208·23-s + 0.575·24-s + 1.77·26-s + 0.919·27-s − 0.626·28-s + 1.70·29-s + 0.825·31-s + 0.926·32-s − 0.114·33-s + 1.89·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 3 | \( 1 + 0.930T + 3T^{2} \) |
| 11 | \( 1 - 0.709T + 11T^{2} \) |
| 13 | \( 1 + 3.91T + 13T^{2} \) |
| 17 | \( 1 + 4.79T + 17T^{2} \) |
| 19 | \( 1 - 2.72T + 19T^{2} \) |
| 29 | \( 1 - 9.19T + 29T^{2} \) |
| 31 | \( 1 - 4.59T + 31T^{2} \) |
| 37 | \( 1 + 0.806T + 37T^{2} \) |
| 41 | \( 1 + 5.49T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 + 3.17T + 47T^{2} \) |
| 53 | \( 1 - 14.0T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 + 4.77T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 - 16.5T + 71T^{2} \) |
| 73 | \( 1 + 8.26T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 - 3.90T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−8.160149560164231733946441042248, −7.52283257653869841083098975992, −6.65283630953195258138075185048, −6.31440212178410693636197337404, −5.18648478685197061950078643377, −4.39715859713809065408367843323, −2.92739321614102200952666124581, −2.29451008468809057988272646706, −0.952174032703480999434929222774, 0,
0.952174032703480999434929222774, 2.29451008468809057988272646706, 2.92739321614102200952666124581, 4.39715859713809065408367843323, 5.18648478685197061950078643377, 6.31440212178410693636197337404, 6.65283630953195258138075185048, 7.52283257653869841083098975992, 8.160149560164231733946441042248
