| L(s) = 1 | − 4·5-s − 7-s + 2·17-s + 2·19-s − 8·23-s + 11·25-s − 2·29-s − 4·31-s + 4·35-s + 6·37-s − 2·41-s + 8·43-s − 4·47-s + 49-s + 10·53-s + 6·59-s + 4·61-s + 12·67-s + 14·73-s − 8·79-s + 6·83-s − 8·85-s + 10·89-s − 8·95-s + 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 0.377·7-s + 0.485·17-s + 0.458·19-s − 1.66·23-s + 11/5·25-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 − 0.583·47-s + 1/7·49-s + 1.37·53-s + 0.781·59-s + 0.512·61-s + 1.46·67-s + 1.63·73-s − 0.900·79-s + 0.658·83-s − 0.867·85-s + 1.05·89-s − 0.820·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.080292482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.080292482\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 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
−13.97222990431042, −13.39244564951105, −12.78391200961547, −12.33823743259466, −11.99074812716799, −11.47651192567626, −11.11342416544822, −10.52085485279694, −9.913450385354867, −9.456142897981295, −8.812259091828228, −8.184502155652071, −7.884598236972474, −7.438109391877251, −6.883777888040665, −6.325249170073147, −5.563243512355917, −5.131292473691784, −4.239722301159349, −3.886840591944981, −3.557468748843864, −2.769819385238413, −2.105931837112192, −1.017555752250232, −0.4012327123789533,
0.4012327123789533, 1.017555752250232, 2.105931837112192, 2.769819385238413, 3.557468748843864, 3.886840591944981, 4.239722301159349, 5.131292473691784, 5.563243512355917, 6.325249170073147, 6.883777888040665, 7.438109391877251, 7.884598236972474, 8.184502155652071, 8.812259091828228, 9.456142897981295, 9.913450385354867, 10.52085485279694, 11.11342416544822, 11.47651192567626, 11.99074812716799, 12.33823743259466, 12.78391200961547, 13.39244564951105, 13.97222990431042
