| L(s) = 1 | + 1.78·2-s + 1.19·4-s + 4.04·7-s − 1.43·8-s − 4.80·11-s − 0.533·13-s + 7.23·14-s − 4.96·16-s − 0.299·17-s − 6.02·19-s − 8.60·22-s + 0.379·23-s − 0.954·26-s + 4.84·28-s + 29-s + 9.14·31-s − 6.00·32-s − 0.536·34-s − 8.51·37-s − 10.7·38-s + 2.24·41-s − 6.01·43-s − 5.76·44-s + 0.677·46-s − 0.299·47-s + 9.37·49-s − 0.639·52-s + ⋯ |
| L(s) = 1 | + 1.26·2-s + 0.599·4-s + 1.52·7-s − 0.506·8-s − 1.45·11-s − 0.147·13-s + 1.93·14-s − 1.24·16-s − 0.0727·17-s − 1.38·19-s − 1.83·22-s + 0.0790·23-s − 0.187·26-s + 0.916·28-s + 0.185·29-s + 1.64·31-s − 1.06·32-s − 0.0919·34-s − 1.40·37-s − 1.74·38-s + 0.349·41-s − 0.916·43-s − 0.868·44-s + 0.0999·46-s − 0.0437·47-s + 1.33·49-s − 0.0886·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 1.78T + 2T^{2} \) |
| 7 | \( 1 - 4.04T + 7T^{2} \) |
| 11 | \( 1 + 4.80T + 11T^{2} \) |
| 13 | \( 1 + 0.533T + 13T^{2} \) |
| 17 | \( 1 + 0.299T + 17T^{2} \) |
| 19 | \( 1 + 6.02T + 19T^{2} \) |
| 23 | \( 1 - 0.379T + 23T^{2} \) |
| 31 | \( 1 - 9.14T + 31T^{2} \) |
| 37 | \( 1 + 8.51T + 37T^{2} \) |
| 41 | \( 1 - 2.24T + 41T^{2} \) |
| 43 | \( 1 + 6.01T + 43T^{2} \) |
| 47 | \( 1 + 0.299T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 + 6.53T + 59T^{2} \) |
| 61 | \( 1 + 0.755T + 61T^{2} \) |
| 67 | \( 1 + 3.95T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 - 7.01T + 79T^{2} \) |
| 83 | \( 1 - 6.15T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 + 2.41T + 97T^{2} \) |
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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
−7.67546821186223778012390350172, −6.77280480824046703207973964211, −6.00101015906205873731253263319, −5.29451776355603120893877325613, −4.63440585042917025899725355580, −4.45495684441609207439052348199, −3.22467703461032169142575892404, −2.50581384106191891068129795134, −1.67480703337893258359266784394, 0,
1.67480703337893258359266784394, 2.50581384106191891068129795134, 3.22467703461032169142575892404, 4.45495684441609207439052348199, 4.63440585042917025899725355580, 5.29451776355603120893877325613, 6.00101015906205873731253263319, 6.77280480824046703207973964211, 7.67546821186223778012390350172
