L(s) = 1 | + (0.888 − 1.10i)2-s + (−0.558 − 0.558i)3-s + (−0.420 − 1.95i)4-s + (2.23 − 0.0162i)5-s + (−1.11 + 0.118i)6-s + (0.707 + 0.707i)7-s + (−2.52 − 1.27i)8-s − 2.37i·9-s + (1.96 − 2.47i)10-s − 0.332·11-s + (−0.856 + 1.32i)12-s + (−0.344 + 0.344i)13-s + (1.40 − 0.149i)14-s + (−1.25 − 1.23i)15-s + (−3.64 + 1.64i)16-s + (2.45 − 2.45i)17-s + ⋯ |
L(s) = 1 | + (0.628 − 0.777i)2-s + (−0.322 − 0.322i)3-s + (−0.210 − 0.977i)4-s + (0.999 − 0.00726i)5-s + (−0.453 + 0.0482i)6-s + (0.267 + 0.267i)7-s + (−0.892 − 0.450i)8-s − 0.792i·9-s + (0.622 − 0.782i)10-s − 0.100·11-s + (−0.247 + 0.382i)12-s + (−0.0954 + 0.0954i)13-s + (0.375 − 0.0399i)14-s + (−0.324 − 0.319i)15-s + (−0.911 + 0.411i)16-s + (0.594 − 0.594i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.226 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.226 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.09254 - 1.37564i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.09254 - 1.37564i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.888 + 1.10i)T \) |
| 5 | \( 1 + (-2.23 + 0.0162i)T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 3 | \( 1 + (0.558 + 0.558i)T + 3iT^{2} \) |
| 11 | \( 1 + 0.332T + 11T^{2} \) |
| 13 | \( 1 + (0.344 - 0.344i)T - 13iT^{2} \) |
| 17 | \( 1 + (-2.45 + 2.45i)T - 17iT^{2} \) |
| 19 | \( 1 - 3.31iT - 19T^{2} \) |
| 23 | \( 1 + (3.67 - 3.67i)T - 23iT^{2} \) |
| 29 | \( 1 + 4.73T + 29T^{2} \) |
| 31 | \( 1 + 6.34iT - 31T^{2} \) |
| 37 | \( 1 + (-3.90 - 3.90i)T + 37iT^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 + (-4.00 - 4.00i)T + 43iT^{2} \) |
| 47 | \( 1 + (-8.92 - 8.92i)T + 47iT^{2} \) |
| 53 | \( 1 + (6.32 - 6.32i)T - 53iT^{2} \) |
| 59 | \( 1 + 1.33iT - 59T^{2} \) |
| 61 | \( 1 - 4.71iT - 61T^{2} \) |
| 67 | \( 1 + (5.65 - 5.65i)T - 67iT^{2} \) |
| 71 | \( 1 + 13.9iT - 71T^{2} \) |
| 73 | \( 1 + (-2.81 - 2.81i)T + 73iT^{2} \) |
| 79 | \( 1 + 9.77T + 79T^{2} \) |
| 83 | \( 1 + (7.54 + 7.54i)T + 83iT^{2} \) |
| 89 | \( 1 + 3.25iT - 89T^{2} \) |
| 97 | \( 1 + (5.14 - 5.14i)T - 97iT^{2} \) |
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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
−11.73798644018577225819322525809, −10.84743267863413136354107191663, −9.653034821004371739799695072049, −9.300003720252837264682325363703, −7.55358307822086248521117470107, −5.99507132590754665492557636432, −5.76417756778991126878137552700, −4.25076360776931252027104740141, −2.75931343444960337638347279521, −1.36308641979149436275259593507,
2.41336529779078443007107016467, 4.13577600582408707121911153227, 5.22516248162183840036578428802, 5.90234229578137556910294252712, 7.09671947142474199501322176939, 8.107183741620705906069949945729, 9.190607941205485870752535411787, 10.33903512233595410867752729170, 11.12231490501379208573473561643, 12.43939919124143072237909513180