| L(s) = 1 | + 1.41·2-s + 1.41·3-s + 3.41·5-s + 2.00·6-s + 3.41·7-s − 2.82·8-s − 0.999·9-s + 4.82·10-s + 1.82·13-s + 4.82·14-s + 4.82·15-s − 4.00·16-s − 2.17·17-s − 1.41·18-s − 0.828·19-s + 4.82·21-s + 6.65·23-s − 4·24-s + 6.65·25-s + 2.58·26-s − 5.65·27-s + 4.24·29-s + 6.82·30-s − 3·31-s + ⋯ |
| L(s) = 1 | + 1.00·2-s + 0.816·3-s + 1.52·5-s + 0.816·6-s + 1.29·7-s − 0.999·8-s − 0.333·9-s + 1.52·10-s + 0.507·13-s + 1.29·14-s + 1.24·15-s − 1.00·16-s − 0.526·17-s − 0.333·18-s − 0.190·19-s + 1.05·21-s + 1.38·23-s − 0.816·24-s + 1.33·25-s + 0.507·26-s − 1.08·27-s + 0.787·29-s + 1.24·30-s − 0.538·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5203 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5203 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.985796480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.985796480\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 - 3.41T + 5T^{2} \) |
| 7 | \( 1 - 3.41T + 7T^{2} \) |
| 13 | \( 1 - 1.82T + 13T^{2} \) |
| 17 | \( 1 + 2.17T + 17T^{2} \) |
| 19 | \( 1 + 0.828T + 19T^{2} \) |
| 23 | \( 1 - 6.65T + 23T^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - 8.48T + 37T^{2} \) |
| 41 | \( 1 + 1.82T + 41T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 + 4.82T + 59T^{2} \) |
| 61 | \( 1 - 0.242T + 61T^{2} \) |
| 67 | \( 1 + 7.48T + 67T^{2} \) |
| 71 | \( 1 + 3.17T + 71T^{2} \) |
| 73 | \( 1 - 16.2T + 73T^{2} \) |
| 79 | \( 1 + 4.82T + 79T^{2} \) |
| 83 | \( 1 + 3.34T + 83T^{2} \) |
| 89 | \( 1 + 1.75T + 89T^{2} \) |
| 97 | \( 1 - 1.82T + 97T^{2} \) |
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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
−8.482994150201153334192416738804, −7.48665188839116686035295880310, −6.48571904259338100382092663016, −5.84445556967167862519089122427, −5.23093222944373225694866503026, −4.64271285667052179047134655690, −3.76059870448052157106312403443, −2.72891316997293665714263652817, −2.28199806851198218034883349739, −1.20446523971811235585971151166,
1.20446523971811235585971151166, 2.28199806851198218034883349739, 2.72891316997293665714263652817, 3.76059870448052157106312403443, 4.64271285667052179047134655690, 5.23093222944373225694866503026, 5.84445556967167862519089122427, 6.48571904259338100382092663016, 7.48665188839116686035295880310, 8.482994150201153334192416738804
