| L(s) = 1 | + 3-s + 4.41·7-s + 9-s + 4.45·11-s − 5.74·13-s + 6.57·17-s − 2.78·19-s + 4.41·21-s + 1.31·23-s + 27-s − 4.84·29-s − 0.197·31-s + 4.45·33-s − 8.20·37-s − 5.74·39-s + 7.84·41-s − 0.412·43-s + 7.79·47-s + 12.4·49-s + 6.57·51-s − 0.315·53-s − 2.78·57-s + 2.51·59-s + 9.33·61-s + 4.41·63-s + 11.9·67-s + 1.31·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.66·7-s + 0.333·9-s + 1.34·11-s − 1.59·13-s + 1.59·17-s − 0.638·19-s + 0.963·21-s + 0.274·23-s + 0.192·27-s − 0.899·29-s − 0.0354·31-s + 0.774·33-s − 1.34·37-s − 0.920·39-s + 1.22·41-s − 0.0629·43-s + 1.13·47-s + 1.78·49-s + 0.920·51-s − 0.0433·53-s − 0.368·57-s + 0.327·59-s + 1.19·61-s + 0.556·63-s + 1.45·67-s + 0.158·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.561868267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.561868267\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4.41T + 7T^{2} \) |
| 11 | \( 1 - 4.45T + 11T^{2} \) |
| 13 | \( 1 + 5.74T + 13T^{2} \) |
| 17 | \( 1 - 6.57T + 17T^{2} \) |
| 19 | \( 1 + 2.78T + 19T^{2} \) |
| 23 | \( 1 - 1.31T + 23T^{2} \) |
| 29 | \( 1 + 4.84T + 29T^{2} \) |
| 31 | \( 1 + 0.197T + 31T^{2} \) |
| 37 | \( 1 + 8.20T + 37T^{2} \) |
| 41 | \( 1 - 7.84T + 41T^{2} \) |
| 43 | \( 1 + 0.412T + 43T^{2} \) |
| 47 | \( 1 - 7.79T + 47T^{2} \) |
| 53 | \( 1 + 0.315T + 53T^{2} \) |
| 59 | \( 1 - 2.51T + 59T^{2} \) |
| 61 | \( 1 - 9.33T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 - 0.509T + 71T^{2} \) |
| 73 | \( 1 - 15.7T + 73T^{2} \) |
| 79 | \( 1 + 3.43T + 79T^{2} \) |
| 83 | \( 1 + 5.37T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 + 5.07T + 97T^{2} \) |
| 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
−7.925070020291660492012394876243, −7.31355152582576558020029633999, −6.75965490233870239172350223283, −5.52088295783275323956406907689, −5.13221508268820210322776360743, −4.22033001048568962087802445025, −3.70469993648128115859381239751, −2.52026190416447036145630156571, −1.83225666429736404406630047377, −0.989569165714878291437276167750,
0.989569165714878291437276167750, 1.83225666429736404406630047377, 2.52026190416447036145630156571, 3.70469993648128115859381239751, 4.22033001048568962087802445025, 5.13221508268820210322776360743, 5.52088295783275323956406907689, 6.75965490233870239172350223283, 7.31355152582576558020029633999, 7.925070020291660492012394876243
