L(s) = 1 | + (1.38 − 0.295i)2-s + 1.66i·3-s + (1.82 − 0.817i)4-s + (2.22 − 0.220i)5-s + (0.491 + 2.29i)6-s + (2.28 − 1.67i)8-s + 0.236·9-s + (3.01 − 0.962i)10-s − 4.81i·11-s + (1.35 + 3.03i)12-s − 2.14·13-s + (0.366 + 3.69i)15-s + (2.66 − 2.98i)16-s + 5.02·17-s + (0.326 − 0.0698i)18-s − 1.36·19-s + ⋯ |
L(s) = 1 | + (0.977 − 0.209i)2-s + 0.959i·3-s + (0.912 − 0.408i)4-s + (0.995 − 0.0986i)5-s + (0.200 + 0.938i)6-s + (0.807 − 0.590i)8-s + 0.0787·9-s + (0.952 − 0.304i)10-s − 1.45i·11-s + (0.392 + 0.875i)12-s − 0.594·13-s + (0.0946 + 0.955i)15-s + (0.665 − 0.746i)16-s + 1.21·17-s + (0.0770 − 0.0164i)18-s − 0.312·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.58802 + 0.176689i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.58802 + 0.176689i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.38 + 0.295i)T \) |
| 5 | \( 1 + (-2.22 + 0.220i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 1.66iT - 3T^{2} \) |
| 11 | \( 1 + 4.81iT - 11T^{2} \) |
| 13 | \( 1 + 2.14T + 13T^{2} \) |
| 17 | \( 1 - 5.02T + 17T^{2} \) |
| 19 | \( 1 + 1.36T + 19T^{2} \) |
| 23 | \( 1 + 5.18T + 23T^{2} \) |
| 29 | \( 1 + 6.43T + 29T^{2} \) |
| 31 | \( 1 + 4.62T + 31T^{2} \) |
| 37 | \( 1 - 9.82iT - 37T^{2} \) |
| 41 | \( 1 - 4.71iT - 41T^{2} \) |
| 43 | \( 1 + 0.141T + 43T^{2} \) |
| 47 | \( 1 - 2.55iT - 47T^{2} \) |
| 53 | \( 1 + 4.84iT - 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 - 10.1iT - 61T^{2} \) |
| 67 | \( 1 - 9.64T + 67T^{2} \) |
| 71 | \( 1 - 9.58iT - 71T^{2} \) |
| 73 | \( 1 + 1.67T + 73T^{2} \) |
| 79 | \( 1 - 11.8iT - 79T^{2} \) |
| 83 | \( 1 - 0.811iT - 83T^{2} \) |
| 89 | \( 1 + 16.0iT - 89T^{2} \) |
| 97 | \( 1 - 1.76T + 97T^{2} \) |
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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
−9.976519102351300494964615980161, −9.670184863883879296452123384477, −8.450628062825298781845394366710, −7.30887767122884159584079868589, −6.12020209320716014670980442669, −5.59577024403653414120460604442, −4.78334224037148177462912763743, −3.73022051128516001545653514066, −2.91765180050905360426082725672, −1.51668331080445335445414894223,
1.79718388630141139602623651953, 2.22032711936415080555945160840, 3.74543591275488929093842542381, 4.89871603092514844148512813012, 5.74200232584618368704561681617, 6.51641195975747280694047204974, 7.46653740517553948206642635565, 7.65160036644781704879301493188, 9.285637756814458982618912966622, 10.05532637526589518617796325310