| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 6·11-s − 2·13-s − 14-s + 16-s + 2·17-s + 4·19-s + 6·22-s + 4·23-s + 2·26-s + 28-s − 2·29-s − 2·31-s − 32-s − 2·34-s + 10·37-s − 4·38-s − 6·41-s + 2·43-s − 6·44-s − 4·46-s − 2·47-s + 49-s − 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 1.80·11-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.917·19-s + 1.27·22-s + 0.834·23-s + 0.392·26-s + 0.188·28-s − 0.371·29-s − 0.359·31-s − 0.176·32-s − 0.342·34-s + 1.64·37-s − 0.648·38-s − 0.937·41-s + 0.304·43-s − 0.904·44-s − 0.589·46-s − 0.291·47-s + 1/7·49-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.187768238590823859567772963200, −7.60289813430143191565826067594, −7.21847849519225090818421577273, −5.97942583043752073588754512516, −5.30260827632870822989734895277, −4.60173005340533618350070882652, −3.16465410230008956815685939833, −2.57078841725378594731422126283, −1.36824692647650071149587985920, 0,
1.36824692647650071149587985920, 2.57078841725378594731422126283, 3.16465410230008956815685939833, 4.60173005340533618350070882652, 5.30260827632870822989734895277, 5.97942583043752073588754512516, 7.21847849519225090818421577273, 7.60289813430143191565826067594, 8.187768238590823859567772963200
