| L(s) = 1 | + 2.44·5-s + 2.55i·7-s + 5.76i·11-s − i·13-s − 1.56i·17-s − 5.82·19-s − 2.69·23-s + 0.978·25-s − 6.76·29-s + 5.29i·31-s + 6.24i·35-s + 1.74i·37-s + 7.99i·41-s − 2.92·43-s + 3.40·47-s + ⋯ |
| L(s) = 1 | + 1.09·5-s + 0.964i·7-s + 1.73i·11-s − 0.277i·13-s − 0.379i·17-s − 1.33·19-s − 0.561·23-s + 0.195·25-s − 1.25·29-s + 0.950i·31-s + 1.05i·35-s + 0.287i·37-s + 1.24i·41-s − 0.446·43-s + 0.497·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.280860450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.280860450\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.44T + 5T^{2} \) |
| 7 | \( 1 - 2.55iT - 7T^{2} \) |
| 11 | \( 1 - 5.76iT - 11T^{2} \) |
| 17 | \( 1 + 1.56iT - 17T^{2} \) |
| 19 | \( 1 + 5.82T + 19T^{2} \) |
| 23 | \( 1 + 2.69T + 23T^{2} \) |
| 29 | \( 1 + 6.76T + 29T^{2} \) |
| 31 | \( 1 - 5.29iT - 31T^{2} \) |
| 37 | \( 1 - 1.74iT - 37T^{2} \) |
| 41 | \( 1 - 7.99iT - 41T^{2} \) |
| 43 | \( 1 + 2.92T + 43T^{2} \) |
| 47 | \( 1 - 3.40T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 + 9.66iT - 59T^{2} \) |
| 61 | \( 1 + 0.282iT - 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 1.05T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 7.55iT - 79T^{2} \) |
| 83 | \( 1 + 14.2iT - 83T^{2} \) |
| 89 | \( 1 + 7.27iT - 89T^{2} \) |
| 97 | \( 1 - 4.07T + 97T^{2} \) |
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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
−9.009326605125087681050384044382, −8.103650902453833195389047256731, −7.30876367224517582533815114399, −6.42778403415454269691266957413, −5.92953491017795600249070383550, −5.03204851171490763729921035304, −4.48310588132780408966353549250, −3.17852767116840771353837418416, −2.07175316292284769934366657971, −1.83769620952538400029828150283,
0.33073833287498347530691463350, 1.58384759999926321023499124230, 2.47813912331195886062872666230, 3.68123096166209865356848177333, 4.18457479221964825950469349653, 5.46416364724060798751738360769, 5.98439415815892910508993288861, 6.54428140521450675616191603900, 7.50753108584556767051426268534, 8.266391444530999362711929707113
