L(s) = 1 | − 3-s − 7-s + 9-s + 11-s + 6·13-s − 2·17-s − 4·19-s + 21-s − 27-s + 2·29-s + 8·31-s − 33-s + 6·37-s − 6·39-s + 10·41-s − 4·43-s + 8·47-s + 49-s + 2·51-s + 6·53-s + 4·57-s − 4·59-s + 10·61-s − 63-s − 12·67-s − 2·73-s − 77-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.66·13-s − 0.485·17-s − 0.917·19-s + 0.218·21-s − 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.174·33-s + 0.986·37-s − 0.960·39-s + 1.56·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s + 0.529·57-s − 0.520·59-s + 1.28·61-s − 0.125·63-s − 1.46·67-s − 0.234·73-s − 0.113·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.946174495\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.946174495\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.41049736754567, −11.97177406454780, −11.70275593926707, −10.99946033319652, −10.70810344933662, −10.51395524340406, −9.758550611458692, −9.328175261773529, −8.894972054587583, −8.380920782462092, −8.033705702505182, −7.406762926079374, −6.717550696426142, −6.485118686159221, −6.002813849071323, −5.751030430545963, −4.958961797951358, −4.364013844898827, −4.086793517504006, −3.569100084911917, −2.833340616089393, −2.344553467792423, −1.627745177877646, −0.8870998946472962, −0.5964649351233395,
0.5964649351233395, 0.8870998946472962, 1.627745177877646, 2.344553467792423, 2.833340616089393, 3.569100084911917, 4.086793517504006, 4.364013844898827, 4.958961797951358, 5.751030430545963, 6.002813849071323, 6.485118686159221, 6.717550696426142, 7.406762926079374, 8.033705702505182, 8.380920782462092, 8.894972054587583, 9.328175261773529, 9.758550611458692, 10.51395524340406, 10.70810344933662, 10.99946033319652, 11.70275593926707, 11.97177406454780, 12.41049736754567