L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 11-s + 2·13-s + 15-s − 6·17-s − 8·19-s + 21-s + 6·23-s + 25-s − 27-s + 6·29-s − 2·31-s − 33-s + 35-s + 2·37-s − 2·39-s − 8·43-s − 45-s + 12·47-s + 49-s + 6·51-s + 6·53-s − 55-s + 8·57-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s + 0.258·15-s − 1.45·17-s − 1.83·19-s + 0.218·21-s + 1.25·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.359·31-s − 0.174·33-s + 0.169·35-s + 0.328·37-s − 0.320·39-s − 1.21·43-s − 0.149·45-s + 1.75·47-s + 1/7·49-s + 0.840·51-s + 0.824·53-s − 0.134·55-s + 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.89293253142216, −15.62334764561781, −15.03589173476844, −14.56964261194533, −13.70132294762686, −13.10783954306313, −12.86461797173223, −12.15879819801911, −11.53589897587417, −11.07101487532579, −10.55648011336931, −10.11718178093809, −9.064608563695911, −8.777771407582531, −8.278857233705826, −7.200166282520569, −6.862215810106555, −6.281952435134070, −5.736750011846418, −4.686892849973117, −4.397413069428199, −3.672811475085687, −2.774541413327988, −1.981835877800762, −0.9199674234004866, 0,
0.9199674234004866, 1.981835877800762, 2.774541413327988, 3.672811475085687, 4.397413069428199, 4.686892849973117, 5.736750011846418, 6.281952435134070, 6.862215810106555, 7.200166282520569, 8.278857233705826, 8.777771407582531, 9.064608563695911, 10.11718178093809, 10.55648011336931, 11.07101487532579, 11.53589897587417, 12.15879819801911, 12.86461797173223, 13.10783954306313, 13.70132294762686, 14.56964261194533, 15.03589173476844, 15.62334764561781, 15.89293253142216