| L(s) = 1 | − 0.317·3-s − 2.89·9-s − 3.78i·11-s + 1.89i·17-s + 5.97i·19-s + 1.87·27-s + 1.20i·33-s − 6.79·41-s + 8.48·43-s + 7·49-s − 0.603i·51-s − 1.89i·57-s + 14.1i·59-s + 16.3·67-s − 15.6i·73-s + ⋯ |
| L(s) = 1 | − 0.183·3-s − 0.966·9-s − 1.14i·11-s + 0.460i·17-s + 1.37i·19-s + 0.360·27-s + 0.209i·33-s − 1.06·41-s + 1.29·43-s + 49-s − 0.0845i·51-s − 0.251i·57-s + 1.84i·59-s + 1.99·67-s − 1.83i·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.350704721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.350704721\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 0.317T + 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 3.78iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 1.89iT - 17T^{2} \) |
| 19 | \( 1 - 5.97iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 6.79T + 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 14.1iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 16.3T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 15.6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 17.0T + 83T^{2} \) |
| 89 | \( 1 + 4.10T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−8.601741694883686603043154626394, −8.143220732547945347364362844363, −7.29233525967086709467625753523, −6.11112508458919345055670173493, −5.92224054908356469652361546858, −5.03050800031715969172751699242, −3.86430474383328412304597001552, −3.23254197082269130964774556293, −2.16203483443736676637964150574, −0.822448319716665190293184541911,
0.58282748740622291405485056933, 2.11608160927965895114788012817, 2.85142967645754438315352807658, 3.95737574808788203148537093993, 4.92348751532406194567968311412, 5.39490057853500975458997152967, 6.49609101104016920933069268026, 7.02305980394499674493907932514, 7.85232747260750437667436887853, 8.668914596305290049484426446045
