L(s) = 1 | + (−0.707 − 0.707i)2-s + (−1.16 + 1.28i)3-s + 1.00i·4-s + (−1.31 + 1.80i)5-s + (1.73 − 0.0807i)6-s + (1.43 − 1.43i)7-s + (0.707 − 0.707i)8-s + (−0.279 − 2.98i)9-s + (2.20 − 0.344i)10-s − 3.03i·11-s + (−1.28 − 1.16i)12-s + (−1.30 − 1.30i)13-s − 2.03·14-s + (−0.774 − 3.79i)15-s − 1.00·16-s + (0.0287 + 0.0287i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (−0.673 + 0.739i)3-s + 0.500i·4-s + (−0.589 + 0.807i)5-s + (0.706 − 0.0329i)6-s + (0.542 − 0.542i)7-s + (0.250 − 0.250i)8-s + (−0.0931 − 0.995i)9-s + (0.698 − 0.108i)10-s − 0.915i·11-s + (−0.369 − 0.336i)12-s + (−0.361 − 0.361i)13-s − 0.542·14-s + (−0.199 − 0.979i)15-s − 0.250·16-s + (0.00696 + 0.00696i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.781799 - 0.139132i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.781799 - 0.139132i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (1.16 - 1.28i)T \) |
| 5 | \( 1 + (1.31 - 1.80i)T \) |
| 23 | \( 1 + (0.707 - 0.707i)T \) |
good | 7 | \( 1 + (-1.43 + 1.43i)T - 7iT^{2} \) |
| 11 | \( 1 + 3.03iT - 11T^{2} \) |
| 13 | \( 1 + (1.30 + 1.30i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.0287 - 0.0287i)T + 17iT^{2} \) |
| 19 | \( 1 - 2.41iT - 19T^{2} \) |
| 29 | \( 1 - 8.32T + 29T^{2} \) |
| 31 | \( 1 - 10.8T + 31T^{2} \) |
| 37 | \( 1 + (0.399 - 0.399i)T - 37iT^{2} \) |
| 41 | \( 1 + 9.45iT - 41T^{2} \) |
| 43 | \( 1 + (0.108 + 0.108i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.341 - 0.341i)T + 47iT^{2} \) |
| 53 | \( 1 + (-0.120 + 0.120i)T - 53iT^{2} \) |
| 59 | \( 1 - 2.10T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 + (6.68 - 6.68i)T - 67iT^{2} \) |
| 71 | \( 1 + 5.69iT - 71T^{2} \) |
| 73 | \( 1 + (-6.43 - 6.43i)T + 73iT^{2} \) |
| 79 | \( 1 + 5.14iT - 79T^{2} \) |
| 83 | \( 1 + (3.07 - 3.07i)T - 83iT^{2} \) |
| 89 | \( 1 - 12.0T + 89T^{2} \) |
| 97 | \( 1 + (-7.38 + 7.38i)T - 97iT^{2} \) |
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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
−10.32548715031889191448538338568, −10.09781349503536349529319537105, −8.684836167935899837128563181619, −7.986592354799843364789856399580, −6.94677673090637124433091362481, −5.99173365096781500360057317348, −4.69605099555453431522946047925, −3.79896971789361312011166003354, −2.84193085485005562189891967114, −0.73084159656145798254700218379,
1.00546426208648743728414424696, 2.31942845918856864561531555742, 4.65518940982763082216011633923, 4.96416689091990257740150688034, 6.26072317170955490071735984235, 7.03892901350444410011476769155, 8.003819598905310857568754382716, 8.456198180671581680145057357776, 9.552868883217658222445561847199, 10.47395787816822725727016800763