| L(s) = 1 | + 20.2·2-s − 99.3·3-s − 101.·4-s + 1.79e3·5-s − 2.01e3·6-s − 1.93e3·7-s − 1.24e4·8-s − 9.82e3·9-s + 3.64e4·10-s − 5.36e4·11-s + 1.00e4·12-s − 3.92e4·14-s − 1.78e5·15-s − 1.99e5·16-s + 3.96e5·17-s − 1.98e5·18-s + 5.03e5·19-s − 1.82e5·20-s + 1.92e5·21-s − 1.08e6·22-s − 1.19e6·23-s + 1.23e6·24-s + 1.28e6·25-s + 2.92e6·27-s + 1.96e5·28-s − 4.89e6·29-s − 3.61e6·30-s + ⋯ |
| L(s) = 1 | + 0.895·2-s − 0.707·3-s − 0.198·4-s + 1.28·5-s − 0.633·6-s − 0.304·7-s − 1.07·8-s − 0.498·9-s + 1.15·10-s − 1.10·11-s + 0.140·12-s − 0.272·14-s − 0.911·15-s − 0.761·16-s + 1.15·17-s − 0.446·18-s + 0.885·19-s − 0.255·20-s + 0.215·21-s − 0.989·22-s − 0.892·23-s + 0.759·24-s + 0.657·25-s + 1.06·27-s + 0.0604·28-s − 1.28·29-s − 0.815·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.033587516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.033587516\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| good | 2 | \( 1 - 20.2T + 512T^{2} \) |
| 3 | \( 1 + 99.3T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.79e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.93e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 5.36e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 3.96e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 5.03e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.19e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.89e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.72e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 6.11e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.63e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.72e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.60e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.91e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 4.85e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.31e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.25e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.00e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.58e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 8.39e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.52e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.14e9T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.39e9T + 7.60e17T^{2} \) |
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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
−11.24714369123142277368708991909, −10.04641278952975979494580988914, −9.391286667206298187367630206189, −7.946510303619309756819105951093, −6.29019846076886358222422232044, −5.57762457991478546004412566836, −5.11846814778327808714810679785, −3.45119323309733544021314295732, −2.34698272331300324783110239448, −0.62411101472687854066428048216,
0.62411101472687854066428048216, 2.34698272331300324783110239448, 3.45119323309733544021314295732, 5.11846814778327808714810679785, 5.57762457991478546004412566836, 6.29019846076886358222422232044, 7.946510303619309756819105951093, 9.391286667206298187367630206189, 10.04641278952975979494580988914, 11.24714369123142277368708991909
