L(s) = 1 | + (1.28 − 1.28i)2-s − 1.28i·4-s + (0.565 + 2.16i)5-s + (0.707 + 0.707i)7-s + (0.918 + 0.918i)8-s + (3.49 + 2.04i)10-s − 4.14i·11-s + (4.57 − 4.57i)13-s + 1.81·14-s + 4.92·16-s + (−5.27 + 5.27i)17-s + 3.06i·19-s + (2.77 − 0.725i)20-s + (−5.30 − 5.30i)22-s + (−3.82 − 3.82i)23-s + ⋯ |
L(s) = 1 | + (0.905 − 0.905i)2-s − 0.641i·4-s + (0.252 + 0.967i)5-s + (0.267 + 0.267i)7-s + (0.324 + 0.324i)8-s + (1.10 + 0.647i)10-s − 1.24i·11-s + (1.26 − 1.26i)13-s + 0.484·14-s + 1.23·16-s + (−1.27 + 1.27i)17-s + 0.702i·19-s + (0.620 − 0.162i)20-s + (−1.13 − 1.13i)22-s + (−0.797 − 0.797i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 + 0.557i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.12456 - 0.647371i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.12456 - 0.647371i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.565 - 2.16i)T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 2 | \( 1 + (-1.28 + 1.28i)T - 2iT^{2} \) |
| 11 | \( 1 + 4.14iT - 11T^{2} \) |
| 13 | \( 1 + (-4.57 + 4.57i)T - 13iT^{2} \) |
| 17 | \( 1 + (5.27 - 5.27i)T - 17iT^{2} \) |
| 19 | \( 1 - 3.06iT - 19T^{2} \) |
| 23 | \( 1 + (3.82 + 3.82i)T + 23iT^{2} \) |
| 29 | \( 1 + 5.24T + 29T^{2} \) |
| 31 | \( 1 + 2.42T + 31T^{2} \) |
| 37 | \( 1 + (0.834 + 0.834i)T + 37iT^{2} \) |
| 41 | \( 1 - 4.19iT - 41T^{2} \) |
| 43 | \( 1 + (-3.71 + 3.71i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.14 + 3.14i)T - 47iT^{2} \) |
| 53 | \( 1 + (4.79 + 4.79i)T + 53iT^{2} \) |
| 59 | \( 1 + 1.97T + 59T^{2} \) |
| 61 | \( 1 - 1.88T + 61T^{2} \) |
| 67 | \( 1 + (10.7 + 10.7i)T + 67iT^{2} \) |
| 71 | \( 1 - 9.76iT - 71T^{2} \) |
| 73 | \( 1 + (-0.978 + 0.978i)T - 73iT^{2} \) |
| 79 | \( 1 + 4.09iT - 79T^{2} \) |
| 83 | \( 1 + (-3.68 - 3.68i)T + 83iT^{2} \) |
| 89 | \( 1 + 7.39T + 89T^{2} \) |
| 97 | \( 1 + (3.07 + 3.07i)T + 97iT^{2} \) |
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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
−11.37049891804343977325556488199, −10.86434108657524460827877813173, −10.35551638211741291249697015478, −8.641644434358633487994245953817, −7.916512005174910719478932628092, −6.19041471946510195586456057662, −5.67740850382860577428410208474, −3.96518657271871246730791506533, −3.24543163931492185495462868481, −1.96386158091869062717575421094,
1.73213572941184700107499969409, 4.15763579166485795153789806696, 4.62990131737119458315530464733, 5.73401774142051687231928941911, 6.80383880669533432677697705132, 7.56280985260195866481161726375, 8.965898965925892841989438321526, 9.578076847903522976661060822774, 11.02320456173078305690572709988, 11.96320983347755402046941054052