L(s) = 1 | + 1.23i·2-s + (0.460 + 1.66i)3-s + 0.473·4-s + (0.5 − 0.866i)5-s + (−2.06 + 0.568i)6-s + (2.47 − 0.930i)7-s + 3.05i·8-s + (−2.57 + 1.53i)9-s + (1.07 + 0.617i)10-s + (−1.10 + 0.635i)11-s + (0.217 + 0.789i)12-s + (1.67 − 0.968i)13-s + (1.15 + 3.06i)14-s + (1.67 + 0.436i)15-s − 2.83·16-s + (0.676 − 1.17i)17-s + ⋯ |
L(s) = 1 | + 0.873i·2-s + (0.265 + 0.964i)3-s + 0.236·4-s + (0.223 − 0.387i)5-s + (−0.842 + 0.232i)6-s + (0.936 − 0.351i)7-s + 1.08i·8-s + (−0.858 + 0.512i)9-s + (0.338 + 0.195i)10-s + (−0.331 + 0.191i)11-s + (0.0628 + 0.228i)12-s + (0.465 − 0.268i)13-s + (0.307 + 0.817i)14-s + (0.432 + 0.112i)15-s − 0.707·16-s + (0.164 − 0.284i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.320 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.00538 + 1.40141i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.00538 + 1.40141i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.460 - 1.66i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-2.47 + 0.930i)T \) |
good | 2 | \( 1 - 1.23iT - 2T^{2} \) |
| 11 | \( 1 + (1.10 - 0.635i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.67 + 0.968i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.676 + 1.17i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.724 - 0.418i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.914 + 0.527i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (8.49 + 4.90i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 2.24iT - 31T^{2} \) |
| 37 | \( 1 + (4.38 + 7.58i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.47 - 6.01i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.63 + 6.30i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 1.69T + 47T^{2} \) |
| 53 | \( 1 + (0.148 + 0.0855i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 9.77T + 59T^{2} \) |
| 61 | \( 1 - 2.84iT - 61T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 - 6.48iT - 71T^{2} \) |
| 73 | \( 1 + (-9.10 - 5.25i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 16.0T + 79T^{2} \) |
| 83 | \( 1 + (-4.56 + 7.90i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (9.41 + 16.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (8.14 + 4.70i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.61559342180645066162352945622, −10.98424157567321345679122496936, −10.07309739295430367753512697233, −8.887125511410220056041949624983, −8.127023337021119327919031744624, −7.31451546678871095284189864294, −5.79190362004463300819844771098, −5.15212352863245545170433957633, −3.97067019266658732690646995274, −2.22156652903463851028744772938,
1.49764573185984965076356241414, 2.43821510309469655871574256284, 3.66998922065598058328366207112, 5.52773943221756159095601567742, 6.59447857353884153614161915522, 7.55076526958771952805336253264, 8.511816041411763793114983364225, 9.598170376273395239054721855954, 10.92024114941432236783134248989, 11.28247313170118377952363721942