L(s) = 1 | + (0.0747 − 0.997i)3-s + (0.365 − 0.930i)7-s + (−0.988 − 0.149i)9-s + (0.0931 + 0.116i)13-s + (−0.365 − 0.632i)19-s + (−0.900 − 0.433i)21-s + (0.365 + 0.930i)25-s + (−0.222 + 0.974i)27-s + (0.733 − 1.26i)31-s + (−1.40 + 1.29i)37-s + (0.123 − 0.0841i)39-s + (1.78 + 0.858i)43-s + (−0.733 − 0.680i)49-s + (−0.658 + 0.317i)57-s + (0.326 − 0.302i)61-s + ⋯ |
L(s) = 1 | + (0.0747 − 0.997i)3-s + (0.365 − 0.930i)7-s + (−0.988 − 0.149i)9-s + (0.0931 + 0.116i)13-s + (−0.365 − 0.632i)19-s + (−0.900 − 0.433i)21-s + (0.365 + 0.930i)25-s + (−0.222 + 0.974i)27-s + (0.733 − 1.26i)31-s + (−1.40 + 1.29i)37-s + (0.123 − 0.0841i)39-s + (1.78 + 0.858i)43-s + (−0.733 − 0.680i)49-s + (−0.658 + 0.317i)57-s + (0.326 − 0.302i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9233181873\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9233181873\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.0747 + 0.997i)T \) |
| 7 | \( 1 + (-0.365 + 0.930i)T \) |
good | 5 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 11 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 13 | \( 1 + (-0.0931 - 0.116i)T + (-0.222 + 0.974i)T^{2} \) |
| 17 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 19 | \( 1 + (0.365 + 0.632i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 29 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 31 | \( 1 + (-0.733 + 1.26i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (1.40 - 1.29i)T + (0.0747 - 0.997i)T^{2} \) |
| 41 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 43 | \( 1 + (-1.78 - 0.858i)T + (0.623 + 0.781i)T^{2} \) |
| 47 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 53 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 59 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 61 | \( 1 + (-0.326 + 0.302i)T + (0.0747 - 0.997i)T^{2} \) |
| 67 | \( 1 + (0.826 - 1.43i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 73 | \( 1 + (-0.698 - 1.77i)T + (-0.733 + 0.680i)T^{2} \) |
| 79 | \( 1 + (-0.733 - 1.26i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 89 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 97 | \( 1 - 1.24T + T^{2} \) |
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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
−10.96690936815417066079464651638, −9.848567644221284712757421374895, −8.777048114440234034271017921023, −7.943774577113719724398871095920, −7.15749228497716721173506811377, −6.43128230434516971850366538328, −5.24153085192507873973227689592, −4.03231416953390946073746124661, −2.67402538236873830691499900714, −1.23546709850735635284275309449,
2.26339410567486130003182245167, 3.46481160600584306661912146577, 4.61654263122742842290714323482, 5.47037861850002388712401376297, 6.35076668042073016767483506656, 7.80905014141108221262079585424, 8.713907011770914578056453870768, 9.198264842554699103185159186660, 10.41199586215820557577401335107, 10.79357082728569519639095271904