L(s) = 1 | + 20.8i·5-s − 13.9i·7-s − 34.5·11-s + 31.3·13-s + 34.4i·17-s − 120. i·19-s + 137.·23-s − 309.·25-s + 93.1i·29-s − 111. i·31-s + 289.·35-s + 146.·37-s − 8.44i·41-s − 427. i·43-s − 318.·47-s + ⋯ |
L(s) = 1 | + 1.86i·5-s − 0.750i·7-s − 0.945·11-s + 0.668·13-s + 0.491i·17-s − 1.45i·19-s + 1.24·23-s − 2.47·25-s + 0.596i·29-s − 0.645i·31-s + 1.40·35-s + 0.651·37-s − 0.0321i·41-s − 1.51i·43-s − 0.989·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.827853808\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.827853808\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 - 20.8iT - 125T^{2} \) |
| 7 | \( 1 + 13.9iT - 343T^{2} \) |
| 11 | \( 1 + 34.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 31.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 34.4iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 120. iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 137.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 93.1iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 111. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 146.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 8.44iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 427. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 318.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 291. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 364.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 289.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 305. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 102.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 442.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 245. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 478.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.41e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.15e3T + 9.12e5T^{2} \) |
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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
−8.984799102806241205640136881458, −7.955137310792556900487829183016, −7.19265601868473088551015363048, −6.78211286574200979643962746309, −5.91207131085554213880675957939, −4.82715579216067296176634939890, −3.66917118454582832268621867934, −3.02347858218736351089257276256, −2.13359299957408491201442718045, −0.52111469394317473248700534776,
0.798828659653351034334979721570, 1.67853455596717881829148258514, 2.91254707080353913066820454707, 4.13311047498426403005458387434, 5.01690051663081129155927555896, 5.50431451164367456843688190141, 6.31787086115885832260441467802, 7.79978838424907229797047668771, 8.163522119374571201073923565353, 8.983337827006868291562106741132