| L(s) = 1 | − 1.52·2-s − 5.66·4-s − 7·7-s + 20.8·8-s − 51.3·11-s + 87.2·13-s + 10.6·14-s + 13.4·16-s + 80.6·17-s − 29.8·19-s + 78.4·22-s + 1.71·23-s − 133.·26-s + 39.6·28-s + 204.·29-s − 150.·31-s − 187.·32-s − 123.·34-s − 366.·37-s + 45.5·38-s − 176.·41-s − 394.·43-s + 290.·44-s − 2.61·46-s + 507.·47-s + 49·49-s − 494.·52-s + ⋯ |
| L(s) = 1 | − 0.540·2-s − 0.708·4-s − 0.377·7-s + 0.922·8-s − 1.40·11-s + 1.86·13-s + 0.204·14-s + 0.209·16-s + 1.15·17-s − 0.360·19-s + 0.760·22-s + 0.0155·23-s − 1.00·26-s + 0.267·28-s + 1.31·29-s − 0.870·31-s − 1.03·32-s − 0.621·34-s − 1.62·37-s + 0.194·38-s − 0.673·41-s − 1.40·43-s + 0.996·44-s − 0.00839·46-s + 1.57·47-s + 0.142·49-s − 1.31·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.019005360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.019005360\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 1.52T + 8T^{2} \) |
| 11 | \( 1 + 51.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 87.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 80.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 29.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 1.71T + 1.21e4T^{2} \) |
| 29 | \( 1 - 204.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 150.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 366.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 176.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 394.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 507.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 149.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 463.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 380.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 797.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 220.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 111.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 853.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 935.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 783.T + 9.12e5T^{2} \) |
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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
−8.868820404449857066703911504674, −8.402349778251486675697884218117, −7.75137278635140754549300163128, −6.73092785286303177362935293017, −5.67732702280076687705137954563, −5.05631853113260372289548557758, −3.86709925316885292600966922847, −3.13728852869349007714694463705, −1.62301793931616179337817228211, −0.55548841134244910379733601227,
0.55548841134244910379733601227, 1.62301793931616179337817228211, 3.13728852869349007714694463705, 3.86709925316885292600966922847, 5.05631853113260372289548557758, 5.67732702280076687705137954563, 6.73092785286303177362935293017, 7.75137278635140754549300163128, 8.402349778251486675697884218117, 8.868820404449857066703911504674
