L(s) = 1 | + (−0.914 − 0.848i)5-s + (0.955 − 0.294i)7-s + (0.826 − 0.563i)9-s + (1.36 + 0.930i)11-s + (0.266 − 0.680i)17-s + (−0.5 + 0.866i)19-s + (0.603 + 1.53i)23-s + (0.0414 + 0.553i)25-s + (−1.12 − 0.541i)35-s + (−0.277 − 1.21i)43-s + (−1.23 − 0.185i)45-s + (−0.109 + 1.46i)47-s + (0.826 − 0.563i)49-s + (−0.458 − 2.00i)55-s + (−0.147 + 0.0222i)61-s + ⋯ |
L(s) = 1 | + (−0.914 − 0.848i)5-s + (0.955 − 0.294i)7-s + (0.826 − 0.563i)9-s + (1.36 + 0.930i)11-s + (0.266 − 0.680i)17-s + (−0.5 + 0.866i)19-s + (0.603 + 1.53i)23-s + (0.0414 + 0.553i)25-s + (−1.12 − 0.541i)35-s + (−0.277 − 1.21i)43-s + (−1.23 − 0.185i)45-s + (−0.109 + 1.46i)47-s + (0.826 − 0.563i)49-s + (−0.458 − 2.00i)55-s + (−0.147 + 0.0222i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 + 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 + 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.436801660\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.436801660\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-0.955 + 0.294i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
good | 3 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 5 | \( 1 + (0.914 + 0.848i)T + (0.0747 + 0.997i)T^{2} \) |
| 11 | \( 1 + (-1.36 - 0.930i)T + (0.365 + 0.930i)T^{2} \) |
| 13 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 17 | \( 1 + (-0.266 + 0.680i)T + (-0.733 - 0.680i)T^{2} \) |
| 23 | \( 1 + (-0.603 - 1.53i)T + (-0.733 + 0.680i)T^{2} \) |
| 29 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 41 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 43 | \( 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2} \) |
| 47 | \( 1 + (0.109 - 1.46i)T + (-0.988 - 0.149i)T^{2} \) |
| 53 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 59 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 61 | \( 1 + (0.147 - 0.0222i)T + (0.955 - 0.294i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 73 | \( 1 + (0.0747 + 0.997i)T + (-0.988 + 0.149i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.658 + 0.317i)T + (0.623 + 0.781i)T^{2} \) |
| 89 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.657896214486806733375150324163, −7.76264084013015765664401300209, −7.34773725621237009815356877787, −6.61389980757906712242798718591, −5.47565556046807491325534399592, −4.58846750587244601167155377933, −4.17497812640936550742101015456, −3.48972103652971538141167284287, −1.73174979884162949069378551601, −1.13607967126861699214337220069,
1.19281341270905322285417476442, 2.34251241415137264576270793660, 3.37626912677244997486844241385, 4.19627721565754872507008786378, 4.74252609466834948633934226515, 5.87812873421350026655562666830, 6.78436043734147236162400233402, 7.12669967909171973597175991034, 8.275052750440290925003953550378, 8.394055954942170782940489904103