| L(s) = 1 | + 3.06·3-s − 5-s + 4.06·7-s + 6.41·9-s + 3.06·11-s − 5.41·13-s − 3.06·15-s + 1.72·17-s − 5.41·19-s + 12.4·21-s + 23-s + 25-s + 10.4·27-s − 7.34·29-s + 8.06·31-s + 9.41·33-s − 4.06·35-s + 0.789·37-s − 16.6·39-s + 11.8·41-s + 4·43-s − 6.41·45-s − 1.44·47-s + 9.55·49-s + 5.27·51-s + 1.48·53-s − 3.06·55-s + ⋯ |
| L(s) = 1 | + 1.77·3-s − 0.447·5-s + 1.53·7-s + 2.13·9-s + 0.925·11-s − 1.50·13-s − 0.792·15-s + 0.417·17-s − 1.24·19-s + 2.72·21-s + 0.208·23-s + 0.200·25-s + 2.01·27-s − 1.36·29-s + 1.44·31-s + 1.63·33-s − 0.687·35-s + 0.129·37-s − 2.66·39-s + 1.85·41-s + 0.609·43-s − 0.956·45-s − 0.210·47-s + 1.36·49-s + 0.739·51-s + 0.204·53-s − 0.413·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.786029052\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.786029052\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 3.06T + 3T^{2} \) |
| 7 | \( 1 - 4.06T + 7T^{2} \) |
| 11 | \( 1 - 3.06T + 11T^{2} \) |
| 13 | \( 1 + 5.41T + 13T^{2} \) |
| 17 | \( 1 - 1.72T + 17T^{2} \) |
| 19 | \( 1 + 5.41T + 19T^{2} \) |
| 29 | \( 1 + 7.34T + 29T^{2} \) |
| 31 | \( 1 - 8.06T + 31T^{2} \) |
| 37 | \( 1 - 0.789T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 1.44T + 47T^{2} \) |
| 53 | \( 1 - 1.48T + 53T^{2} \) |
| 59 | \( 1 - 9.62T + 59T^{2} \) |
| 61 | \( 1 - 12.0T + 61T^{2} \) |
| 67 | \( 1 + 0.514T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 9.48T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 + 5.20T + 97T^{2} \) |
| 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
−7.951880438863534862016404790023, −7.43624114896869028226293513631, −6.96898248403881146757499617588, −5.75129753444436849903924326463, −4.58563489838552559514187022662, −4.37561726328273381545100169882, −3.57677614078158981341791483956, −2.49613833969055787595444455903, −2.09745186725326121953888859052, −1.07785166957767567754086829764,
1.07785166957767567754086829764, 2.09745186725326121953888859052, 2.49613833969055787595444455903, 3.57677614078158981341791483956, 4.37561726328273381545100169882, 4.58563489838552559514187022662, 5.75129753444436849903924326463, 6.96898248403881146757499617588, 7.43624114896869028226293513631, 7.951880438863534862016404790023
