L(s) = 1 | − 3-s + 3.77·5-s + 1.75·7-s + 9-s − 2.41·11-s − 1.58·13-s − 3.77·15-s + 0.811·17-s − 3.41·19-s − 1.75·21-s − 6.93·23-s + 9.26·25-s − 27-s − 0.520·29-s + 3.53·31-s + 2.41·33-s + 6.62·35-s − 9.20·37-s + 1.58·39-s − 3.31·41-s − 6.00·43-s + 3.77·45-s − 12.4·47-s − 3.92·49-s − 0.811·51-s + 12.1·53-s − 9.11·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.68·5-s + 0.662·7-s + 0.333·9-s − 0.728·11-s − 0.438·13-s − 0.975·15-s + 0.196·17-s − 0.784·19-s − 0.382·21-s − 1.44·23-s + 1.85·25-s − 0.192·27-s − 0.0966·29-s + 0.635·31-s + 0.420·33-s + 1.11·35-s − 1.51·37-s + 0.253·39-s − 0.517·41-s − 0.914·43-s + 0.563·45-s − 1.81·47-s − 0.560·49-s − 0.113·51-s + 1.66·53-s − 1.22·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 167 | \( 1 + T \) |
good | 5 | \( 1 - 3.77T + 5T^{2} \) |
| 7 | \( 1 - 1.75T + 7T^{2} \) |
| 11 | \( 1 + 2.41T + 11T^{2} \) |
| 13 | \( 1 + 1.58T + 13T^{2} \) |
| 17 | \( 1 - 0.811T + 17T^{2} \) |
| 19 | \( 1 + 3.41T + 19T^{2} \) |
| 23 | \( 1 + 6.93T + 23T^{2} \) |
| 29 | \( 1 + 0.520T + 29T^{2} \) |
| 31 | \( 1 - 3.53T + 31T^{2} \) |
| 37 | \( 1 + 9.20T + 37T^{2} \) |
| 41 | \( 1 + 3.31T + 41T^{2} \) |
| 43 | \( 1 + 6.00T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 73 | \( 1 - 3.71T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 - 0.129T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−7.38311095962233292690446801267, −6.58151298841874019712310793719, −6.10091220687478917087719750778, −5.24547937048540050264540025332, −5.08771766893199557541076940571, −4.07831821398508420092681198935, −2.86619356586590032616469429539, −2.00531465039884007892947710431, −1.53408474268995673574896956732, 0,
1.53408474268995673574896956732, 2.00531465039884007892947710431, 2.86619356586590032616469429539, 4.07831821398508420092681198935, 5.08771766893199557541076940571, 5.24547937048540050264540025332, 6.10091220687478917087719750778, 6.58151298841874019712310793719, 7.38311095962233292690446801267