| L(s) = 1 | − 2-s − 2.81·3-s + 4-s + 2.81·6-s + 7-s − 8-s + 4.91·9-s + 11-s − 2.81·12-s + 4.91·13-s − 14-s + 16-s − 4.81·17-s − 4.91·18-s + 1.28·19-s − 2.81·21-s − 22-s − 7.49·23-s + 2.81·24-s − 4.91·26-s − 5.39·27-s + 28-s − 1.86·29-s + 9.15·31-s − 32-s − 2.81·33-s + 4.81·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.62·3-s + 0.5·4-s + 1.14·6-s + 0.377·7-s − 0.353·8-s + 1.63·9-s + 0.301·11-s − 0.812·12-s + 1.36·13-s − 0.267·14-s + 0.250·16-s − 1.16·17-s − 1.15·18-s + 0.295·19-s − 0.613·21-s − 0.213·22-s − 1.56·23-s + 0.574·24-s − 0.964·26-s − 1.03·27-s + 0.188·28-s − 0.346·29-s + 1.64·31-s − 0.176·32-s − 0.489·33-s + 0.825·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2.81T + 3T^{2} \) |
| 13 | \( 1 - 4.91T + 13T^{2} \) |
| 17 | \( 1 + 4.81T + 17T^{2} \) |
| 19 | \( 1 - 1.28T + 19T^{2} \) |
| 23 | \( 1 + 7.49T + 23T^{2} \) |
| 29 | \( 1 + 1.86T + 29T^{2} \) |
| 31 | \( 1 - 9.15T + 31T^{2} \) |
| 37 | \( 1 + 6.75T + 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 - 2.62T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 - 9.74T + 53T^{2} \) |
| 59 | \( 1 - 1.39T + 59T^{2} \) |
| 61 | \( 1 - 6.57T + 61T^{2} \) |
| 67 | \( 1 + 3.62T + 67T^{2} \) |
| 71 | \( 1 + 0.338T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 + 6.28T + 79T^{2} \) |
| 83 | \( 1 + 9.75T + 83T^{2} \) |
| 89 | \( 1 + 6.71T + 89T^{2} \) |
| 97 | \( 1 - 0.338T + 97T^{2} \) |
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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
−8.356136701802513467243647672800, −7.18391285317034497987918116237, −6.52283607133239097393371611613, −6.10119309957758216442933188847, −5.28498293156046986337320886554, −4.45037095920173891272128342375, −3.58016061360294288777868936929, −2.01379437227274366053493331175, −1.14615192743331857624804780453, 0,
1.14615192743331857624804780453, 2.01379437227274366053493331175, 3.58016061360294288777868936929, 4.45037095920173891272128342375, 5.28498293156046986337320886554, 6.10119309957758216442933188847, 6.52283607133239097393371611613, 7.18391285317034497987918116237, 8.356136701802513467243647672800
