| L(s) = 1 | − 2·3-s − 4·5-s + 7-s + 9-s + 8·15-s − 2·17-s − 2·21-s + 8·23-s + 11·25-s + 4·27-s − 2·29-s − 4·31-s − 4·35-s + 6·37-s + 2·41-s + 8·43-s − 4·45-s − 4·47-s + 49-s + 4·51-s + 10·53-s − 6·59-s + 4·61-s + 63-s + 12·67-s − 16·69-s − 14·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.78·5-s + 0.377·7-s + 1/3·9-s + 2.06·15-s − 0.485·17-s − 0.436·21-s + 1.66·23-s + 11/5·25-s + 0.769·27-s − 0.371·29-s − 0.718·31-s − 0.676·35-s + 0.986·37-s + 0.312·41-s + 1.21·43-s − 0.596·45-s − 0.583·47-s + 1/7·49-s + 0.560·51-s + 1.37·53-s − 0.781·59-s + 0.512·61-s + 0.125·63-s + 1.46·67-s − 1.92·69-s − 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7738679515\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7738679515\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−15.78108158381509, −15.13376063516763, −14.76700261351364, −14.21393879157704, −13.14803199736177, −12.81072419745435, −12.20164732096089, −11.65692246477082, −11.27132798304633, −10.94112522915804, −10.49444275228607, −9.437823400910320, −8.804989450773321, −8.324383415335091, −7.515586756347223, −7.221600365435100, −6.558436676475240, −5.794962777770801, −5.087856667142559, −4.612717630490790, −4.013319185875995, −3.291501565731642, −2.461152809923141, −1.090006199790433, −0.4740113624676991,
0.4740113624676991, 1.090006199790433, 2.461152809923141, 3.291501565731642, 4.013319185875995, 4.612717630490790, 5.087856667142559, 5.794962777770801, 6.558436676475240, 7.221600365435100, 7.515586756347223, 8.324383415335091, 8.804989450773321, 9.437823400910320, 10.49444275228607, 10.94112522915804, 11.27132798304633, 11.65692246477082, 12.20164732096089, 12.81072419745435, 13.14803199736177, 14.21393879157704, 14.76700261351364, 15.13376063516763, 15.78108158381509
