L(s) = 1 | + 11.5i·5-s + 23.0i·7-s − 1.19i·11-s + 58.3·13-s + (54.8 − 43.6i)17-s + 138.·19-s − 180. i·23-s − 7.98·25-s − 115. i·29-s − 210. i·31-s − 265.·35-s − 210. i·37-s + 297. i·41-s + 174.·43-s + 199.·47-s + ⋯ |
L(s) = 1 | + 1.03i·5-s + 1.24i·7-s − 0.0327i·11-s + 1.24·13-s + (0.782 − 0.622i)17-s + 1.67·19-s − 1.64i·23-s − 0.0638·25-s − 0.736i·29-s − 1.22i·31-s − 1.28·35-s − 0.936i·37-s + 1.13i·41-s + 0.619·43-s + 0.619·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.627151602\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.627151602\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( 1 + (-54.8 + 43.6i)T \) |
good | 5 | \( 1 - 11.5iT - 125T^{2} \) |
| 7 | \( 1 - 23.0iT - 343T^{2} \) |
| 11 | \( 1 + 1.19iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 58.3T + 2.19e3T^{2} \) |
| 19 | \( 1 - 138.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 180. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 115. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 210. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 210. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 297. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 174.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 199.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 706.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 182.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 489. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 167.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 818. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 1.09e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 110. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 118.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 400.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 924. iT - 9.12e5T^{2} \) |
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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
−9.423864259930650485542204253219, −8.667272167174874226302762067854, −7.79032891960988645613170409501, −6.93745727969605041633160027467, −5.96940611841168790844624814355, −5.52196803437913045541751117751, −4.13038366588530833994395340708, −3.00856938939188668875190301774, −2.44069226292818984970349821845, −0.862789063891583828395430442126,
1.02775468068570632538468626041, 1.30064357202631963396373009876, 3.37712952458858604108434571938, 3.89891505956146213982075263716, 5.10637153719649900276980449015, 5.68383413319315299463644465709, 6.96669639382415972096522784555, 7.61424491837691362980203735914, 8.485653013725259611381357039146, 9.204019810822036482857348123968