| L(s) = 1 | + 5.48·2-s − 3·3-s + 22.0·4-s + 13.3·5-s − 16.4·6-s + 21.4·7-s + 77.3·8-s + 9·9-s + 73.0·10-s − 19.0·11-s − 66.2·12-s + 117.·14-s − 39.9·15-s + 247.·16-s − 71.7·17-s + 49.3·18-s − 102.·19-s + 294.·20-s − 64.2·21-s − 104.·22-s − 37.8·23-s − 231.·24-s + 52.3·25-s − 27·27-s + 473.·28-s + 40.8·29-s − 219.·30-s + ⋯ |
| L(s) = 1 | + 1.93·2-s − 0.577·3-s + 2.76·4-s + 1.19·5-s − 1.11·6-s + 1.15·7-s + 3.41·8-s + 0.333·9-s + 2.31·10-s − 0.522·11-s − 1.59·12-s + 2.24·14-s − 0.687·15-s + 3.86·16-s − 1.02·17-s + 0.646·18-s − 1.23·19-s + 3.28·20-s − 0.667·21-s − 1.01·22-s − 0.342·23-s − 1.97·24-s + 0.419·25-s − 0.192·27-s + 3.19·28-s + 0.261·29-s − 1.33·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.548437566\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.548437566\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 5.48T + 8T^{2} \) |
| 5 | \( 1 - 13.3T + 125T^{2} \) |
| 7 | \( 1 - 21.4T + 343T^{2} \) |
| 11 | \( 1 + 19.0T + 1.33e3T^{2} \) |
| 17 | \( 1 + 71.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 102.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 37.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 40.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 6.05T + 2.97e4T^{2} \) |
| 37 | \( 1 + 285.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 342.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 306.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 346.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 398.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 208.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 546.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 678.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 957.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 270.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.03e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.06e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 427.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 698.T + 9.12e5T^{2} \) |
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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
−10.77581433500513557371336717202, −10.25022712955548870724423315301, −8.524313098501586587409865988977, −7.23257198658050375447433628754, −6.37259872544480970061364198869, −5.59483357643270939597657210505, −4.92471613333518493882894345957, −4.09284915137118379865660329019, −2.40512302146340762385658018354, −1.74109481815853732590797028763,
1.74109481815853732590797028763, 2.40512302146340762385658018354, 4.09284915137118379865660329019, 4.92471613333518493882894345957, 5.59483357643270939597657210505, 6.37259872544480970061364198869, 7.23257198658050375447433628754, 8.524313098501586587409865988977, 10.25022712955548870724423315301, 10.77581433500513557371336717202
