L(s) = 1 | − 1.22i·2-s + 0.804i·3-s + 0.507·4-s + (−2.23 + 0.0991i)5-s + 0.982·6-s − 3.79i·7-s − 3.06i·8-s + 2.35·9-s + (0.121 + 2.72i)10-s − 1.23·11-s + 0.408i·12-s − 3.45i·13-s − 4.63·14-s + (−0.0797 − 1.79i)15-s − 2.72·16-s − 3.00i·17-s + ⋯ |
L(s) = 1 | − 0.863i·2-s + 0.464i·3-s + 0.253·4-s + (−0.999 + 0.0443i)5-s + 0.401·6-s − 1.43i·7-s − 1.08i·8-s + 0.784·9-s + (0.0382 + 0.862i)10-s − 0.373·11-s + 0.117i·12-s − 0.958i·13-s − 1.23·14-s + (−0.0205 − 0.463i)15-s − 0.681·16-s − 0.729i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0443i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.189815687\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.189815687\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (2.23 - 0.0991i)T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + 1.22iT - 2T^{2} \) |
| 3 | \( 1 - 0.804iT - 3T^{2} \) |
| 7 | \( 1 + 3.79iT - 7T^{2} \) |
| 11 | \( 1 + 1.23T + 11T^{2} \) |
| 13 | \( 1 + 3.45iT - 13T^{2} \) |
| 17 | \( 1 + 3.00iT - 17T^{2} \) |
| 23 | \( 1 - 6.14iT - 23T^{2} \) |
| 29 | \( 1 + 4.28T + 29T^{2} \) |
| 31 | \( 1 + 5.10T + 31T^{2} \) |
| 37 | \( 1 - 9.13iT - 37T^{2} \) |
| 41 | \( 1 + 5.33T + 41T^{2} \) |
| 43 | \( 1 + 9.12iT - 43T^{2} \) |
| 47 | \( 1 + 7.30iT - 47T^{2} \) |
| 53 | \( 1 - 3.33iT - 53T^{2} \) |
| 59 | \( 1 + 0.817T + 59T^{2} \) |
| 61 | \( 1 - 6.40T + 61T^{2} \) |
| 67 | \( 1 + 1.01iT - 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 + 11.0iT - 73T^{2} \) |
| 79 | \( 1 + 1.38T + 79T^{2} \) |
| 83 | \( 1 + 2.52iT - 83T^{2} \) |
| 89 | \( 1 - 2.69T + 89T^{2} \) |
| 97 | \( 1 - 3.06iT - 97T^{2} \) |
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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
−9.145121840685712580193028077468, −7.88158994609028337312479228693, −7.29402524631420414470762446002, −6.91077099065764646551775601474, −5.35524455147041262612717117609, −4.40097258219362561788925878690, −3.62138275419632419628486399885, −3.19742538996373936021269983574, −1.57213199680127658974877007125, −0.43000810504227161654835886566,
1.78109993109591185695643448457, 2.62714022672426592092935888238, 4.00677120572285102050179425948, 4.94016240597667533267574742071, 5.87387491981648813075061634517, 6.58262860124378396989519294465, 7.27474947911620170430420169044, 7.958028499633441138436039909630, 8.613377408619180267284933552822, 9.268628796903461369169562678691