| L(s) = 1 | + 7.99·3-s + 5·5-s − 4.09·7-s + 36.8·9-s + 47.4·11-s − 7.45·13-s + 39.9·15-s + 58.5·17-s − 112.·19-s − 32.7·21-s − 23·23-s + 25·25-s + 78.7·27-s + 51.2·29-s + 124.·31-s + 379.·33-s − 20.4·35-s + 366.·37-s − 59.5·39-s − 339.·41-s + 497.·43-s + 184.·45-s + 609.·47-s − 326.·49-s + 468.·51-s + 120.·53-s + 237.·55-s + ⋯ |
| L(s) = 1 | + 1.53·3-s + 0.447·5-s − 0.221·7-s + 1.36·9-s + 1.30·11-s − 0.159·13-s + 0.687·15-s + 0.835·17-s − 1.35·19-s − 0.340·21-s − 0.208·23-s + 0.200·25-s + 0.560·27-s + 0.327·29-s + 0.719·31-s + 2.00·33-s − 0.0988·35-s + 1.62·37-s − 0.244·39-s − 1.29·41-s + 1.76·43-s + 0.610·45-s + 1.89·47-s − 0.951·49-s + 1.28·51-s + 0.312·53-s + 0.581·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.937082771\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.937082771\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 - 7.99T + 27T^{2} \) |
| 7 | \( 1 + 4.09T + 343T^{2} \) |
| 11 | \( 1 - 47.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 7.45T + 2.19e3T^{2} \) |
| 17 | \( 1 - 58.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 112.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 51.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 124.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 366.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 339.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 497.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 609.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 120.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 309.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 76.7T + 2.26e5T^{2} \) |
| 67 | \( 1 + 502.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 347.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 99.0T + 3.89e5T^{2} \) |
| 79 | \( 1 - 990.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 71.3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 914.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 258.T + 9.12e5T^{2} \) |
| 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
−8.879471121926660588168839496321, −8.285213995473342983715050776603, −7.46645603365083264948618116679, −6.58812273400100175455403009219, −5.85709412702418736906481786567, −4.43647815028406431017084884389, −3.82271866289630784398822653776, −2.84117441667152086562684941080, −2.08109274646298116680751870412, −1.01669220867529933462426184054,
1.01669220867529933462426184054, 2.08109274646298116680751870412, 2.84117441667152086562684941080, 3.82271866289630784398822653776, 4.43647815028406431017084884389, 5.85709412702418736906481786567, 6.58812273400100175455403009219, 7.46645603365083264948618116679, 8.285213995473342983715050776603, 8.879471121926660588168839496321
