L(s) = 1 | − 1.66·5-s − 3.57i·7-s + 1.02·11-s + (1.87 + 3.07i)13-s + 5.05·17-s − 1.16·19-s + 8.17·23-s − 2.23·25-s − 4.29i·29-s + 7.98i·31-s + 5.93i·35-s − 9.83·37-s + 1.62i·41-s − 2.35i·43-s − 7.73i·47-s + ⋯ |
L(s) = 1 | − 0.743·5-s − 1.35i·7-s + 0.309·11-s + (0.520 + 0.853i)13-s + 1.22·17-s − 0.266·19-s + 1.70·23-s − 0.447·25-s − 0.798i·29-s + 1.43i·31-s + 1.00i·35-s − 1.61·37-s + 0.253i·41-s − 0.358i·43-s − 1.12i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.759 + 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.759 + 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.690673051\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.690673051\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (-1.87 - 3.07i)T \) |
good | 5 | \( 1 + 1.66T + 5T^{2} \) |
| 7 | \( 1 + 3.57iT - 7T^{2} \) |
| 11 | \( 1 - 1.02T + 11T^{2} \) |
| 17 | \( 1 - 5.05T + 17T^{2} \) |
| 19 | \( 1 + 1.16T + 19T^{2} \) |
| 23 | \( 1 - 8.17T + 23T^{2} \) |
| 29 | \( 1 + 4.29iT - 29T^{2} \) |
| 31 | \( 1 - 7.98iT - 31T^{2} \) |
| 37 | \( 1 + 9.83T + 37T^{2} \) |
| 41 | \( 1 - 1.62iT - 41T^{2} \) |
| 43 | \( 1 + 2.35iT - 43T^{2} \) |
| 47 | \( 1 + 7.73iT - 47T^{2} \) |
| 53 | \( 1 - 11.2iT - 53T^{2} \) |
| 59 | \( 1 - 9.73T + 59T^{2} \) |
| 61 | \( 1 - 11.4iT - 61T^{2} \) |
| 67 | \( 1 - 7.23T + 67T^{2} \) |
| 71 | \( 1 + 10.5iT - 71T^{2} \) |
| 73 | \( 1 + 4.41iT - 73T^{2} \) |
| 79 | \( 1 - 6.94T + 79T^{2} \) |
| 83 | \( 1 - 2.29T + 83T^{2} \) |
| 89 | \( 1 + 9.02iT - 89T^{2} \) |
| 97 | \( 1 + 11.5iT - 97T^{2} \) |
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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
−8.479806431177178489767937178650, −7.44345637594362224723428756054, −7.17673055269094092452710094106, −6.40749520936154516617623724622, −5.31718556628422382440566790034, −4.46405386486623401654670002054, −3.76324636030395756338200969433, −3.22011799096765414869080217929, −1.63021477994219260485274585972, −0.69990825632321635383724216256,
0.884477197281367230673619617988, 2.19386539093542436478710640580, 3.23010645435965284113120057786, 3.73574412915091951344471676120, 5.06458438880574488804810503495, 5.46815941106871076065201859963, 6.32982835448984770882712997154, 7.18702148410809813138001599009, 8.039997590049777508319800050052, 8.472842583956739678712124616853