L(s) = 1 | + (1.22 + 0.707i)2-s + (0.5 + 2.59i)7-s − 2.82i·8-s + (1.22 − 0.707i)11-s − 5.19i·13-s + (−1.22 + 3.53i)14-s + (2.00 − 3.46i)16-s + (−2.44 − 4.24i)17-s + (1.5 + 0.866i)19-s + 2·22-s + (4.89 + 2.82i)23-s + (3.67 − 6.36i)26-s − 2.82i·29-s + (1.5 − 0.866i)31-s − 6.92i·34-s + ⋯ |
L(s) = 1 | + (0.866 + 0.499i)2-s + (0.188 + 0.981i)7-s − 0.999i·8-s + (0.369 − 0.213i)11-s − 1.44i·13-s + (−0.327 + 0.944i)14-s + (0.500 − 0.866i)16-s + (−0.594 − 1.02i)17-s + (0.344 + 0.198i)19-s + 0.426·22-s + (1.02 + 0.589i)23-s + (0.720 − 1.24i)26-s − 0.525i·29-s + (0.269 − 0.155i)31-s − 1.18i·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.574738329\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.574738329\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.5 - 2.59i)T \) |
good | 2 | \( 1 + (-1.22 - 0.707i)T + (1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (-1.22 + 0.707i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 5.19iT - 13T^{2} \) |
| 17 | \( 1 + (2.44 + 4.24i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.5 - 0.866i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.89 - 2.82i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 + (-1.5 + 0.866i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 7.34T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 + (-6.12 + 10.6i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.44 + 1.41i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.44 - 4.24i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3 + 1.73i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.5 - 9.52i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 7.07iT - 71T^{2} \) |
| 73 | \( 1 + (1.5 - 0.866i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.5 - 4.33i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 7.34T + 83T^{2} \) |
| 89 | \( 1 + (2.44 - 4.24i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 10.3iT - 97T^{2} \) |
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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
−9.334753578459741482344392392050, −8.606595248536923579830273493998, −7.59030484938915385757859625195, −6.82106290275388085134360760601, −5.78215465332680794027871270825, −5.43534464080181039957380194063, −4.59180089087665299869702336813, −3.46584389713184676935119273367, −2.56481308501511236256083756647, −0.835641293230984037778465550312,
1.41187875042519474911564691397, 2.54516796254332030829960636223, 3.75118067228510691583069313338, 4.30855247604749914596445666907, 4.94082391546738616133489776249, 6.21516372780240403803184983138, 6.96518641361328406338314365978, 7.81785430796280695951084438815, 8.798032041933823790860918276945, 9.378157799421283204778563152037