| L(s) = 1 | + 20·3-s − 74·5-s + 24·7-s + 157·9-s − 478·13-s − 1.48e3·15-s + 1.19e3·17-s − 3.04e3·19-s + 480·21-s + 184·23-s + 2.35e3·25-s − 1.72e3·27-s + 3.28e3·29-s − 5.72e3·31-s − 1.77e3·35-s + 1.03e4·37-s − 9.56e3·39-s + 8.88e3·41-s + 9.18e3·43-s − 1.16e4·45-s + 2.36e4·47-s − 1.62e4·49-s + 2.39e4·51-s + 1.16e4·53-s − 6.08e4·57-s + 1.68e4·59-s + 1.84e4·61-s + ⋯ |
| L(s) = 1 | + 1.28·3-s − 1.32·5-s + 0.185·7-s + 0.646·9-s − 0.784·13-s − 1.69·15-s + 1.00·17-s − 1.93·19-s + 0.237·21-s + 0.0725·23-s + 0.752·25-s − 0.454·27-s + 0.724·29-s − 1.07·31-s − 0.245·35-s + 1.24·37-s − 1.00·39-s + 0.825·41-s + 0.757·43-s − 0.855·45-s + 1.56·47-s − 0.965·49-s + 1.28·51-s + 0.571·53-s − 2.48·57-s + 0.631·59-s + 0.635·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.178313104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.178313104\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 20 T + p^{5} T^{2} \) |
| 5 | \( 1 + 74 T + p^{5} T^{2} \) |
| 7 | \( 1 - 24 T + p^{5} T^{2} \) |
| 13 | \( 1 + 478 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1198 T + p^{5} T^{2} \) |
| 19 | \( 1 + 3044 T + p^{5} T^{2} \) |
| 23 | \( 1 - 8 p T + p^{5} T^{2} \) |
| 29 | \( 1 - 3282 T + p^{5} T^{2} \) |
| 31 | \( 1 + 5728 T + p^{5} T^{2} \) |
| 37 | \( 1 - 10326 T + p^{5} T^{2} \) |
| 41 | \( 1 - 8886 T + p^{5} T^{2} \) |
| 43 | \( 1 - 9188 T + p^{5} T^{2} \) |
| 47 | \( 1 - 23664 T + p^{5} T^{2} \) |
| 53 | \( 1 - 11686 T + p^{5} T^{2} \) |
| 59 | \( 1 - 16876 T + p^{5} T^{2} \) |
| 61 | \( 1 - 18482 T + p^{5} T^{2} \) |
| 67 | \( 1 + 15532 T + p^{5} T^{2} \) |
| 71 | \( 1 + 31960 T + p^{5} T^{2} \) |
| 73 | \( 1 - 4886 T + p^{5} T^{2} \) |
| 79 | \( 1 + 44560 T + p^{5} T^{2} \) |
| 83 | \( 1 + 67364 T + p^{5} T^{2} \) |
| 89 | \( 1 - 71994 T + p^{5} T^{2} \) |
| 97 | \( 1 - 48866 T + p^{5} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.010635101310588421621301538932, −8.395125580608155677663480268830, −7.71653794515254995632555798728, −7.20743690225384077652628964764, −5.86840714341464870208565285107, −4.48252304954047073880658811531, −3.92406949825182913894406903359, −2.94713428351413042023704562300, −2.10421025070901190290208953773, −0.58556448274292676348212056807,
0.58556448274292676348212056807, 2.10421025070901190290208953773, 2.94713428351413042023704562300, 3.92406949825182913894406903359, 4.48252304954047073880658811531, 5.86840714341464870208565285107, 7.20743690225384077652628964764, 7.71653794515254995632555798728, 8.395125580608155677663480268830, 9.010635101310588421621301538932
