| L(s) = 1 | + 1.82i·5-s − 7-s + 3.75i·11-s − 3.09i·13-s + 3.61·17-s + 1.65i·19-s + 7.47·23-s + 1.65·25-s + 2.38i·29-s − 8.97·31-s − 1.82i·35-s + 8.94i·37-s + 3.04·41-s + 1.82i·43-s − 6.21·47-s + ⋯ |
| L(s) = 1 | + 0.818i·5-s − 0.377·7-s + 1.13i·11-s − 0.859i·13-s + 0.875·17-s + 0.379i·19-s + 1.55·23-s + 0.330·25-s + 0.443i·29-s − 1.61·31-s − 0.309i·35-s + 1.47i·37-s + 0.475·41-s + 0.278i·43-s − 0.906·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.276 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.658256951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.658256951\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 1.82iT - 5T^{2} \) |
| 11 | \( 1 - 3.75iT - 11T^{2} \) |
| 13 | \( 1 + 3.09iT - 13T^{2} \) |
| 17 | \( 1 - 3.61T + 17T^{2} \) |
| 19 | \( 1 - 1.65iT - 19T^{2} \) |
| 23 | \( 1 - 7.47T + 23T^{2} \) |
| 29 | \( 1 - 2.38iT - 29T^{2} \) |
| 31 | \( 1 + 8.97T + 31T^{2} \) |
| 37 | \( 1 - 8.94iT - 37T^{2} \) |
| 41 | \( 1 - 3.04T + 41T^{2} \) |
| 43 | \( 1 - 1.82iT - 43T^{2} \) |
| 47 | \( 1 + 6.21T + 47T^{2} \) |
| 53 | \( 1 - 3.21iT - 53T^{2} \) |
| 59 | \( 1 + 12.4iT - 59T^{2} \) |
| 61 | \( 1 + 9.91iT - 61T^{2} \) |
| 67 | \( 1 - 8.73iT - 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 9.79T + 73T^{2} \) |
| 79 | \( 1 - 7.24T + 79T^{2} \) |
| 83 | \( 1 + 8.99iT - 83T^{2} \) |
| 89 | \( 1 + 1.54T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−8.087101512870289052546446884041, −7.52349248577946819083826049951, −6.86144782573567280178733298503, −6.35543244028139857945088565879, −5.28365859351057296640860990785, −4.87933935994531735050003739063, −3.55646867076674268441125337456, −3.19646704610014132501225204900, −2.23693248029198153051762184132, −1.10790007300401326905211615306,
0.48290587574069238841802296696, 1.36383036764963356335438841150, 2.56989264338843614601398301409, 3.47084082843913386412505418374, 4.13749975208916856494644038455, 5.18446441256388821388054658616, 5.53499817747521111930780903849, 6.47599156666257120870386588089, 7.14686105742084660080557559563, 7.898419205043455973759638280269
