| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 11-s + 6·13-s + 15-s − 7·17-s + 5·19-s − 21-s − 23-s + 25-s − 27-s + 5·29-s − 8·31-s + 33-s − 35-s + 2·37-s − 6·39-s + 12·41-s + 11·43-s − 45-s + 8·47-s + 49-s + 7·51-s + 11·53-s + 55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.66·13-s + 0.258·15-s − 1.69·17-s + 1.14·19-s − 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.928·29-s − 1.43·31-s + 0.174·33-s − 0.169·35-s + 0.328·37-s − 0.960·39-s + 1.87·41-s + 1.67·43-s − 0.149·45-s + 1.16·47-s + 1/7·49-s + 0.980·51-s + 1.51·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.345312812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.345312812\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−14.02143921237352, −13.60143377313610, −12.96347670486645, −12.69763024810030, −11.96014436320870, −11.48995447256505, −11.02441937009561, −10.80369126322072, −10.29600910631017, −9.334250248123616, −9.033770558469519, −8.556244880148325, −7.844193706923539, −7.418175870938455, −6.879136658972168, −6.186903183752456, −5.786559304730244, −5.243761682003622, −4.455806413002612, −4.078919307964966, −3.551911731084355, −2.620136615489313, −2.052207426116230, −1.039808145870532, −0.6419937465847056,
0.6419937465847056, 1.039808145870532, 2.052207426116230, 2.620136615489313, 3.551911731084355, 4.078919307964966, 4.455806413002612, 5.243761682003622, 5.786559304730244, 6.186903183752456, 6.879136658972168, 7.418175870938455, 7.844193706923539, 8.556244880148325, 9.033770558469519, 9.334250248123616, 10.29600910631017, 10.80369126322072, 11.02441937009561, 11.48995447256505, 11.96014436320870, 12.69763024810030, 12.96347670486645, 13.60143377313610, 14.02143921237352
