L(s) = 1 | − 5-s + 7-s + 4·11-s + 2·13-s + 2·17-s − 4·19-s + 23-s + 25-s − 2·29-s − 35-s + 6·37-s + 6·41-s − 8·43-s − 4·47-s + 49-s − 2·53-s − 4·55-s − 12·59-s − 2·61-s − 2·65-s + 8·67-s − 8·71-s − 2·73-s + 4·77-s + 8·79-s − 8·83-s − 2·85-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.377·7-s + 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s + 0.208·23-s + 1/5·25-s − 0.371·29-s − 0.169·35-s + 0.986·37-s + 0.937·41-s − 1.21·43-s − 0.583·47-s + 1/7·49-s − 0.274·53-s − 0.539·55-s − 1.56·59-s − 0.256·61-s − 0.248·65-s + 0.977·67-s − 0.949·71-s − 0.234·73-s + 0.455·77-s + 0.900·79-s − 0.878·83-s − 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 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
−14.48698745724142, −14.36362430957284, −13.53077240020262, −13.13507696133769, −12.47462617646805, −12.09607347243925, −11.45140143535713, −11.17593933276008, −10.63453749882240, −9.990769846847403, −9.331640991677163, −8.982439424925636, −8.334358462918718, −7.903315337248402, −7.349328670905123, −6.592186761810732, −6.288540915649463, −5.642478544930912, −4.844174295885174, −4.347798664020980, −3.789152670009212, −3.260809339225523, −2.427529130812446, −1.567843618690109, −1.074451361696645, 0,
1.074451361696645, 1.567843618690109, 2.427529130812446, 3.260809339225523, 3.789152670009212, 4.347798664020980, 4.844174295885174, 5.642478544930912, 6.288540915649463, 6.592186761810732, 7.349328670905123, 7.903315337248402, 8.334358462918718, 8.982439424925636, 9.331640991677163, 9.990769846847403, 10.63453749882240, 11.17593933276008, 11.45140143535713, 12.09607347243925, 12.47462617646805, 13.13507696133769, 13.53077240020262, 14.36362430957284, 14.48698745724142