L(s) = 1 | + (0.809 − 0.587i)2-s + (−0.809 − 0.587i)3-s + (0.309 − 0.951i)4-s − 5-s − 6-s + (−0.186 + 0.573i)7-s + (−0.309 − 0.951i)8-s + (0.309 + 0.951i)9-s + (−0.809 + 0.587i)10-s + (−1.99 + 6.15i)11-s + (−0.809 + 0.587i)12-s + (0.974 + 0.708i)13-s + (0.186 + 0.573i)14-s + (0.809 + 0.587i)15-s + (−0.809 − 0.587i)16-s + (1.33 + 4.11i)17-s + ⋯ |
L(s) = 1 | + (0.572 − 0.415i)2-s + (−0.467 − 0.339i)3-s + (0.154 − 0.475i)4-s − 0.447·5-s − 0.408·6-s + (−0.0703 + 0.216i)7-s + (−0.109 − 0.336i)8-s + (0.103 + 0.317i)9-s + (−0.255 + 0.185i)10-s + (−0.602 + 1.85i)11-s + (−0.233 + 0.169i)12-s + (0.270 + 0.196i)13-s + (0.0497 + 0.153i)14-s + (0.208 + 0.151i)15-s + (−0.202 − 0.146i)16-s + (0.324 + 0.997i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.29568 + 0.431283i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.29568 + 0.431283i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 3 | \( 1 + (0.809 + 0.587i)T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + (-1.72 + 5.29i)T \) |
good | 7 | \( 1 + (0.186 - 0.573i)T + (-5.66 - 4.11i)T^{2} \) |
| 11 | \( 1 + (1.99 - 6.15i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-0.974 - 0.708i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.33 - 4.11i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.153 + 0.111i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.56 - 4.81i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-6.79 + 4.93i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + 6.74T + 37T^{2} \) |
| 41 | \( 1 + (2.61 - 1.89i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (9.45 - 6.86i)T + (13.2 - 40.8i)T^{2} \) |
| 47 | \( 1 + (-8.10 - 5.88i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (1.00 + 3.08i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-6.96 - 5.06i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 - 0.666T + 61T^{2} \) |
| 67 | \( 1 + 5.98T + 67T^{2} \) |
| 71 | \( 1 + (0.550 + 1.69i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (1.94 - 5.98i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-2.63 - 8.10i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.585 + 0.425i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (4.97 - 15.3i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-4.10 + 12.6i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.20717252443470533361909468917, −9.676539633345882427774493972245, −8.337724251151585689347249923941, −7.48419872872704195243721597551, −6.68448287193657773729329010617, −5.71074690682348926925656708204, −4.77697083781511193066338464371, −4.01043664223901737804001865937, −2.63174345676919822035924269928, −1.49962371620769368390457206544,
0.58953110164667405379417074222, 2.97610683957910164950265735923, 3.61211227028479354174752588806, 4.91601910637468500103534704809, 5.45250549614384748448942709404, 6.50896886617966386048919820071, 7.21821895283707455911274281532, 8.471366188052098354209272071724, 8.739099675508344533990940017404, 10.42686211568424206777096959735