| L(s) = 1 | + 0.743·5-s − 1.72i·7-s + 4.66i·11-s − i·13-s − 6.58i·17-s + 3.02·19-s − 7.02·23-s − 4.44·25-s − 10.3·29-s + 0.619i·31-s − 1.27i·35-s − 0.689i·37-s − 5.53i·41-s − 8.98·43-s + 0.183·47-s + ⋯ |
| L(s) = 1 | + 0.332·5-s − 0.650i·7-s + 1.40i·11-s − 0.277i·13-s − 1.59i·17-s + 0.694·19-s − 1.46·23-s − 0.889·25-s − 1.92·29-s + 0.111i·31-s − 0.216i·35-s − 0.113i·37-s − 0.863i·41-s − 1.37·43-s + 0.0268·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 + 0.470i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.882 + 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6032331793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6032331793\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| good | 5 | \( 1 - 0.743T + 5T^{2} \) |
| 7 | \( 1 + 1.72iT - 7T^{2} \) |
| 11 | \( 1 - 4.66iT - 11T^{2} \) |
| 17 | \( 1 + 6.58iT - 17T^{2} \) |
| 19 | \( 1 - 3.02T + 19T^{2} \) |
| 23 | \( 1 + 7.02T + 23T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 - 0.619iT - 31T^{2} \) |
| 37 | \( 1 + 0.689iT - 37T^{2} \) |
| 41 | \( 1 + 5.53iT - 41T^{2} \) |
| 43 | \( 1 + 8.98T + 43T^{2} \) |
| 47 | \( 1 - 0.183T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 13.2iT - 59T^{2} \) |
| 61 | \( 1 + 13.1iT - 61T^{2} \) |
| 67 | \( 1 + 0.268T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 2.74T + 73T^{2} \) |
| 79 | \( 1 + 11.0iT - 79T^{2} \) |
| 83 | \( 1 - 6.15iT - 83T^{2} \) |
| 89 | \( 1 - 13.1iT - 89T^{2} \) |
| 97 | \( 1 + 17.2T + 97T^{2} \) |
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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
−8.022719879710425179244187950404, −7.25593240580942699932042680383, −7.09592597496816931538089154553, −5.82582784839384060139237031561, −5.24884202428413044860694916996, −4.33063270134168360413015295642, −3.63386249100021432935277068181, −2.43861643791720205164756021476, −1.63935798989195224856962313434, −0.16366685192399530136192189425,
1.49001231999900356789875510660, 2.32339234506745783058976427940, 3.50620650039057922545680401059, 4.00854495077197658311291745892, 5.37533222447462395284056705582, 5.86491286986530813584991453333, 6.31281189001710313803379791766, 7.47755820413440962469960004771, 8.236521542813833716077772531264, 8.690342563994159326836849501778
