L(s) = 1 | + 1.41·2-s + (−2.53 − 1.59i)3-s + 2.00·4-s + (−3.59 − 2.26i)6-s + 2.64i·7-s + 2.82·8-s + (3.89 + 8.11i)9-s − 7.44i·11-s + (−5.07 − 3.19i)12-s + 13.9i·13-s + 3.74i·14-s + 4.00·16-s + 17.8·17-s + (5.50 + 11.4i)18-s − 33.6·19-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.846 − 0.532i)3-s + 0.500·4-s + (−0.598 − 0.376i)6-s + 0.377i·7-s + 0.353·8-s + (0.432 + 0.901i)9-s − 0.677i·11-s + (−0.423 − 0.266i)12-s + 1.07i·13-s + 0.267i·14-s + 0.250·16-s + 1.05·17-s + (0.305 + 0.637i)18-s − 1.77·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0980 - 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0980 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.451941123\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.451941123\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 + (2.53 + 1.59i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
good | 11 | \( 1 + 7.44iT - 121T^{2} \) |
| 13 | \( 1 - 13.9iT - 169T^{2} \) |
| 17 | \( 1 - 17.8T + 289T^{2} \) |
| 19 | \( 1 + 33.6T + 361T^{2} \) |
| 23 | \( 1 + 7.94T + 529T^{2} \) |
| 29 | \( 1 - 8.60iT - 841T^{2} \) |
| 31 | \( 1 + 56.7T + 961T^{2} \) |
| 37 | \( 1 - 12.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 75.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 60.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 67.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 49.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 82.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 6.51T + 3.72e3T^{2} \) |
| 67 | \( 1 + 27.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 1.41iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 46.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 49.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 42.4T + 6.88e3T^{2} \) |
| 89 | \( 1 + 63.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 34.3iT - 9.40e3T^{2} \) |
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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
−10.19185399170679960573133084618, −9.032310968234492642198623430468, −8.071471314752249439314178032020, −7.16609667919166236900732076253, −6.27374482687982934115146725246, −5.79836947782558303691033124418, −4.78665794252246889156334692474, −3.87642114703993698462415317868, −2.47376675232360784749777851785, −1.37924328787969669584809327186,
0.39241525725608213944348232940, 2.05665398896396758984322172793, 3.61516645241098780950361499730, 4.17815741576048845052321075991, 5.33379717130904595116138013530, 5.77241703280239172369778200272, 6.88321766499441501482622289011, 7.56232227036808988988385089887, 8.763713082578128750276795267565, 9.860299740417430204069239419678