L(s) = 1 | + 2·3-s + 7-s + 9-s − 6·11-s − 13-s + 3·17-s − 4·19-s + 2·21-s − 23-s − 4·27-s + 6·29-s − 8·31-s − 12·33-s + 11·37-s − 2·39-s − 8·43-s + 3·47-s + 49-s + 6·51-s + 9·53-s − 8·57-s − 8·61-s + 63-s − 8·67-s − 2·69-s + 6·71-s − 2·73-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.80·11-s − 0.277·13-s + 0.727·17-s − 0.917·19-s + 0.436·21-s − 0.208·23-s − 0.769·27-s + 1.11·29-s − 1.43·31-s − 2.08·33-s + 1.80·37-s − 0.320·39-s − 1.21·43-s + 0.437·47-s + 1/7·49-s + 0.840·51-s + 1.23·53-s − 1.05·57-s − 1.02·61-s + 0.125·63-s − 0.977·67-s − 0.240·69-s + 0.712·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−13.08178399811520, −12.74113824509217, −12.26627774006197, −11.67843966062374, −11.13338867145571, −10.66682019632247, −10.16208298769478, −9.959158566805569, −9.117519442275751, −8.943565786078503, −8.117097340537116, −8.095313729853776, −7.624354981411144, −7.161675194545270, −6.462165055720429, −5.722700414876851, −5.518386776189306, −4.742742904074514, −4.429364661492448, −3.657194951055140, −3.162330556332362, −2.643891042375542, −2.243113470299544, −1.739674601046517, −0.7728114533748104, 0,
0.7728114533748104, 1.739674601046517, 2.243113470299544, 2.643891042375542, 3.162330556332362, 3.657194951055140, 4.429364661492448, 4.742742904074514, 5.518386776189306, 5.722700414876851, 6.462165055720429, 7.161675194545270, 7.624354981411144, 8.095313729853776, 8.117097340537116, 8.943565786078503, 9.117519442275751, 9.959158566805569, 10.16208298769478, 10.66682019632247, 11.13338867145571, 11.67843966062374, 12.26627774006197, 12.74113824509217, 13.08178399811520