| L(s) = 1 | + 5-s + 3·7-s − 11-s + 6·17-s + 4·19-s + 4·23-s + 25-s + 7·31-s + 3·35-s + 10·37-s + 7·43-s + 2·49-s + 2·53-s − 55-s − 15·61-s + 13·67-s − 6·71-s + 3·73-s − 3·77-s − 13·79-s − 6·83-s + 6·85-s + 4·95-s + 9·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.13·7-s − 0.301·11-s + 1.45·17-s + 0.917·19-s + 0.834·23-s + 1/5·25-s + 1.25·31-s + 0.507·35-s + 1.64·37-s + 1.06·43-s + 2/7·49-s + 0.274·53-s − 0.134·55-s − 1.92·61-s + 1.58·67-s − 0.712·71-s + 0.351·73-s − 0.341·77-s − 1.46·79-s − 0.658·83-s + 0.650·85-s + 0.410·95-s + 0.913·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.362153458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.362153458\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−12.57112274162299, −12.13049163321957, −11.52471596815184, −11.38590279855687, −10.77350403857756, −10.27442562199725, −9.929892814656036, −9.362702862098907, −9.051324070663794, −8.272498204116246, −8.027801670625545, −7.551080062212638, −7.199216559073011, −6.482726468582375, −5.880386570575987, −5.588068859312704, −5.027789866722716, −4.628619585146319, −4.130276942365386, −3.330646180897386, −2.850869692220443, −2.434104148111033, −1.567275211768284, −1.156072370447394, −0.6753829918474777,
0.6753829918474777, 1.156072370447394, 1.567275211768284, 2.434104148111033, 2.850869692220443, 3.330646180897386, 4.130276942365386, 4.628619585146319, 5.027789866722716, 5.588068859312704, 5.880386570575987, 6.482726468582375, 7.199216559073011, 7.551080062212638, 8.027801670625545, 8.272498204116246, 9.051324070663794, 9.362702862098907, 9.929892814656036, 10.27442562199725, 10.77350403857756, 11.38590279855687, 11.52471596815184, 12.13049163321957, 12.57112274162299
