L(s) = 1 | + (1.73 + 1.73i)5-s + 1.03i·7-s + (−0.896 − 0.896i)11-s + (3.73 − 3.73i)13-s + 3.46·17-s + (−0.896 + 0.896i)19-s − 6.69i·23-s + 0.999i·25-s + (1.73 − 1.73i)29-s + 5.65·31-s + (−1.79 + 1.79i)35-s + (−0.267 − 0.267i)37-s − 6.92i·41-s + (5.79 + 5.79i)43-s − 9.79·47-s + ⋯ |
L(s) = 1 | + (0.774 + 0.774i)5-s + 0.391i·7-s + (−0.270 − 0.270i)11-s + (1.03 − 1.03i)13-s + 0.840·17-s + (−0.205 + 0.205i)19-s − 1.39i·23-s + 0.199i·25-s + (0.321 − 0.321i)29-s + 1.01·31-s + (−0.303 + 0.303i)35-s + (−0.0440 − 0.0440i)37-s − 1.08i·41-s + (0.883 + 0.883i)43-s − 1.42·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.241415551\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.241415551\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.73 - 1.73i)T + 5iT^{2} \) |
| 7 | \( 1 - 1.03iT - 7T^{2} \) |
| 11 | \( 1 + (0.896 + 0.896i)T + 11iT^{2} \) |
| 13 | \( 1 + (-3.73 + 3.73i)T - 13iT^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + (0.896 - 0.896i)T - 19iT^{2} \) |
| 23 | \( 1 + 6.69iT - 23T^{2} \) |
| 29 | \( 1 + (-1.73 + 1.73i)T - 29iT^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + (0.267 + 0.267i)T + 37iT^{2} \) |
| 41 | \( 1 + 6.92iT - 41T^{2} \) |
| 43 | \( 1 + (-5.79 - 5.79i)T + 43iT^{2} \) |
| 47 | \( 1 + 9.79T + 47T^{2} \) |
| 53 | \( 1 + (-4.26 - 4.26i)T + 53iT^{2} \) |
| 59 | \( 1 + (-7.58 - 7.58i)T + 59iT^{2} \) |
| 61 | \( 1 + (0.267 - 0.267i)T - 61iT^{2} \) |
| 67 | \( 1 + (2.96 - 2.96i)T - 67iT^{2} \) |
| 71 | \( 1 + 6.69iT - 71T^{2} \) |
| 73 | \( 1 - 9.46iT - 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + (-5.79 + 5.79i)T - 83iT^{2} \) |
| 89 | \( 1 - 9.46iT - 89T^{2} \) |
| 97 | \( 1 + 3.46T + 97T^{2} \) |
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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
−8.886648293824552382985552415688, −8.334786668748822503129838230390, −7.53278102095712115194437639598, −6.42632399206530761157370335397, −6.01795455927185798770358667834, −5.29284030042017261038380918453, −4.08765906429071738360159484903, −2.99639131378623184676355123768, −2.42871798848988770539288551347, −0.967557709309647747140302927788,
1.10613615609081171528611004594, 1.88370099653177804765636106435, 3.26214843003152329647291203926, 4.20775019955312893239183436766, 5.07358175450648868997954133896, 5.79975728229415331424646645169, 6.60496964447460320818705416900, 7.45390424498652074935380113252, 8.358118932300819104975852205703, 9.005100928618818267101193299626