| L(s) = 1 | + 1.45·2-s + 0.115·4-s − 3.82·5-s + 3.26·7-s − 2.74·8-s − 5.56·10-s + 11-s + 5.15·13-s + 4.74·14-s − 4.21·16-s − 1.81·17-s − 6.25·19-s − 0.441·20-s + 1.45·22-s + 3.26·23-s + 9.63·25-s + 7.49·26-s + 0.376·28-s − 9.11·29-s + 9.69·31-s − 0.651·32-s − 2.63·34-s − 12.4·35-s − 11.3·37-s − 9.09·38-s + 10.4·40-s + 1.74·41-s + ⋯ |
| L(s) = 1 | + 1.02·2-s + 0.0576·4-s − 1.71·5-s + 1.23·7-s − 0.969·8-s − 1.75·10-s + 0.301·11-s + 1.42·13-s + 1.26·14-s − 1.05·16-s − 0.439·17-s − 1.43·19-s − 0.0986·20-s + 0.310·22-s + 0.680·23-s + 1.92·25-s + 1.47·26-s + 0.0710·28-s − 1.69·29-s + 1.74·31-s − 0.115·32-s − 0.452·34-s − 2.10·35-s − 1.87·37-s − 1.47·38-s + 1.65·40-s + 0.272·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.162688066\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.162688066\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 1.45T + 2T^{2} \) |
| 5 | \( 1 + 3.82T + 5T^{2} \) |
| 7 | \( 1 - 3.26T + 7T^{2} \) |
| 13 | \( 1 - 5.15T + 13T^{2} \) |
| 17 | \( 1 + 1.81T + 17T^{2} \) |
| 19 | \( 1 + 6.25T + 19T^{2} \) |
| 23 | \( 1 - 3.26T + 23T^{2} \) |
| 29 | \( 1 + 9.11T + 29T^{2} \) |
| 31 | \( 1 - 9.69T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 - 1.74T + 41T^{2} \) |
| 43 | \( 1 - 1.82T + 43T^{2} \) |
| 47 | \( 1 - 3.66T + 47T^{2} \) |
| 53 | \( 1 - 6.16T + 53T^{2} \) |
| 59 | \( 1 + 8.95T + 59T^{2} \) |
| 67 | \( 1 - 4.57T + 67T^{2} \) |
| 71 | \( 1 - 5.54T + 71T^{2} \) |
| 73 | \( 1 + 7.91T + 73T^{2} \) |
| 79 | \( 1 - 6.29T + 79T^{2} \) |
| 83 | \( 1 - 2.81T + 83T^{2} \) |
| 89 | \( 1 + 5.92T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−8.150612106479933183036901422458, −7.34841243707338218639626018856, −6.57494455465296439167811651559, −5.82129097960144182464588037247, −4.84274264135946992329236262950, −4.42141326087383183962078380589, −3.82103252727906368362317812662, −3.27564515269601645835128740276, −1.97648518939867116796645875303, −0.66377306104887448439206327491,
0.66377306104887448439206327491, 1.97648518939867116796645875303, 3.27564515269601645835128740276, 3.82103252727906368362317812662, 4.42141326087383183962078380589, 4.84274264135946992329236262950, 5.82129097960144182464588037247, 6.57494455465296439167811651559, 7.34841243707338218639626018856, 8.150612106479933183036901422458
