| L(s) = 1 | + 2·3-s + 4·5-s + 7-s + 9-s + 4·11-s − 4·13-s + 8·15-s − 6·17-s + 2·19-s + 2·21-s + 23-s + 11·25-s − 4·27-s + 6·29-s + 8·31-s + 8·33-s + 4·35-s − 10·37-s − 8·39-s + 6·41-s + 12·43-s + 4·45-s + 8·47-s + 49-s − 12·51-s + 2·53-s + 16·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.78·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s − 1.10·13-s + 2.06·15-s − 1.45·17-s + 0.458·19-s + 0.436·21-s + 0.208·23-s + 11/5·25-s − 0.769·27-s + 1.11·29-s + 1.43·31-s + 1.39·33-s + 0.676·35-s − 1.64·37-s − 1.28·39-s + 0.937·41-s + 1.82·43-s + 0.596·45-s + 1.16·47-s + 1/7·49-s − 1.68·51-s + 0.274·53-s + 2.15·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.238981048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.238981048\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−16.94916166131735, −15.80363151491862, −15.39812798426748, −14.47886158428375, −14.29954859611894, −13.86652191017955, −13.46732589662521, −12.68823486033446, −12.08337992637110, −11.36680773072988, −10.48217142183772, −10.04216481397411, −9.343906207038869, −8.887347873131547, −8.700367340589003, −7.557821015779846, −6.990787752681660, −6.272474935407953, −5.717506796755399, −4.753034264609928, −4.282448275224803, −3.062778595310250, −2.472890569720785, −1.982458075272381, −1.069330501498045,
1.069330501498045, 1.982458075272381, 2.472890569720785, 3.062778595310250, 4.282448275224803, 4.753034264609928, 5.717506796755399, 6.272474935407953, 6.990787752681660, 7.557821015779846, 8.700367340589003, 8.887347873131547, 9.343906207038869, 10.04216481397411, 10.48217142183772, 11.36680773072988, 12.08337992637110, 12.68823486033446, 13.46732589662521, 13.86652191017955, 14.29954859611894, 14.47886158428375, 15.39812798426748, 15.80363151491862, 16.94916166131735
