L(s) = 1 | + (−0.826 − 0.563i)2-s + (−0.997 − 0.925i)3-s + (0.365 + 0.930i)4-s + (0.302 + 1.32i)6-s + (0.974 + 0.222i)7-s + (0.222 − 0.974i)8-s + (0.0635 + 0.848i)9-s + (−0.0841 + 1.12i)11-s + (0.496 − 1.26i)12-s + (1.48 − 0.716i)13-s + (−0.680 − 0.733i)14-s + (−0.733 + 0.680i)16-s + (−0.988 + 0.149i)17-s + (0.425 − 0.736i)18-s + (−0.766 − 1.12i)21-s + (0.702 − 0.880i)22-s + ⋯ |
L(s) = 1 | + (−0.826 − 0.563i)2-s + (−0.997 − 0.925i)3-s + (0.365 + 0.930i)4-s + (0.302 + 1.32i)6-s + (0.974 + 0.222i)7-s + (0.222 − 0.974i)8-s + (0.0635 + 0.848i)9-s + (−0.0841 + 1.12i)11-s + (0.496 − 1.26i)12-s + (1.48 − 0.716i)13-s + (−0.680 − 0.733i)14-s + (−0.733 + 0.680i)16-s + (−0.988 + 0.149i)17-s + (0.425 − 0.736i)18-s + (−0.766 − 1.12i)21-s + (0.702 − 0.880i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6871667625\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6871667625\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.826 + 0.563i)T \) |
| 7 | \( 1 + (-0.974 - 0.222i)T \) |
| 17 | \( 1 + (0.988 - 0.149i)T \) |
good | 3 | \( 1 + (0.997 + 0.925i)T + (0.0747 + 0.997i)T^{2} \) |
| 5 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 11 | \( 1 + (0.0841 - 1.12i)T + (-0.988 - 0.149i)T^{2} \) |
| 13 | \( 1 + (-1.48 + 0.716i)T + (0.623 - 0.781i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.858 - 0.129i)T + (0.955 + 0.294i)T^{2} \) |
| 29 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 31 | \( 1 + (0.974 - 1.68i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 41 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 43 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 47 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 53 | \( 1 + (-0.698 - 1.77i)T + (-0.733 + 0.680i)T^{2} \) |
| 59 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 61 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.367 - 0.460i)T + (-0.222 - 0.974i)T^{2} \) |
| 73 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 79 | \( 1 + (-0.997 - 1.72i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 89 | \( 1 + (-0.109 - 1.46i)T + (-0.988 + 0.149i)T^{2} \) |
| 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
−8.649059902684114477405934109052, −8.092596164706765649935201470398, −7.09489641885821884883580791834, −6.86117787210148242290715027570, −5.83346150391146038446882588573, −4.98538956853289925183770161165, −4.04412857388394656513096891107, −2.78718756952811722226016625516, −1.69633659702278923578541159927, −1.09895376787393161775256216249,
0.78403436064583527412418920883, 2.01107999415447734150929260852, 3.62659830537685171480210945874, 4.53834125789914560905395495492, 5.21810361843325993482173046195, 5.91868716172173508576143866859, 6.51279258841133665688157733622, 7.40878988605289469422696899339, 8.356687446892023947957248068065, 8.865147302321086401007807923064