L(s) = 1 | + (0.707 + 0.707i)2-s + (−0.707 − 0.707i)3-s + 1.00i·4-s + (1.19 − 1.88i)5-s − 1.00i·6-s + (0.510 − 2.59i)7-s + (−0.707 + 0.707i)8-s + 1.00i·9-s + (2.18 − 0.489i)10-s + 4.79·11-s + (0.707 − 0.707i)12-s + (0.585 + 0.585i)13-s + (2.19 − 1.47i)14-s + (−2.18 + 0.489i)15-s − 1.00·16-s + (−4.10 + 4.10i)17-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (−0.408 − 0.408i)3-s + 0.500i·4-s + (0.535 − 0.844i)5-s − 0.408i·6-s + (0.192 − 0.981i)7-s + (−0.250 + 0.250i)8-s + 0.333i·9-s + (0.689 − 0.154i)10-s + 1.44·11-s + (0.204 − 0.204i)12-s + (0.162 + 0.162i)13-s + (0.587 − 0.394i)14-s + (−0.563 + 0.126i)15-s − 0.250·16-s + (−0.995 + 0.995i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.203i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.50704 - 0.155221i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.50704 - 0.155221i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (-1.19 + 1.88i)T \) |
| 7 | \( 1 + (-0.510 + 2.59i)T \) |
good | 11 | \( 1 - 4.79T + 11T^{2} \) |
| 13 | \( 1 + (-0.585 - 0.585i)T + 13iT^{2} \) |
| 17 | \( 1 + (4.10 - 4.10i)T - 17iT^{2} \) |
| 19 | \( 1 + 2.36T + 19T^{2} \) |
| 23 | \( 1 + (-2.97 + 2.97i)T - 23iT^{2} \) |
| 29 | \( 1 - 9.94iT - 29T^{2} \) |
| 31 | \( 1 + 3.02iT - 31T^{2} \) |
| 37 | \( 1 + (6.10 + 6.10i)T + 37iT^{2} \) |
| 41 | \( 1 - 10.9iT - 41T^{2} \) |
| 43 | \( 1 + (5.74 - 5.74i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.363 + 0.363i)T - 47iT^{2} \) |
| 53 | \( 1 + (-2.36 + 2.36i)T - 53iT^{2} \) |
| 59 | \( 1 + 2.07T + 59T^{2} \) |
| 61 | \( 1 - 5.55iT - 61T^{2} \) |
| 67 | \( 1 + (0.979 + 0.979i)T + 67iT^{2} \) |
| 71 | \( 1 - 5.25T + 71T^{2} \) |
| 73 | \( 1 + (-6.11 - 6.11i)T + 73iT^{2} \) |
| 79 | \( 1 - 5.10iT - 79T^{2} \) |
| 83 | \( 1 + (3.22 + 3.22i)T + 83iT^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 + (-8.05 + 8.05i)T - 97iT^{2} \) |
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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
−12.68783165805148074475871880864, −11.52707580121522739707516715821, −10.57263465879277589305692942192, −9.141667590251107116941742814791, −8.317504352647498040801339402788, −6.87001730275334379943825160374, −6.30468670130691745772430278441, −4.89091187026500027028296423469, −3.97739024724358775341280596756, −1.51127531557405934118958132210,
2.14154726512495463862268014475, 3.56317202800352766520120139633, 4.96664064393783025382385248973, 6.08357376596615012386997862876, 6.86657312359812776297186687532, 8.864470491037098176096885498928, 9.573660821346382307841723429040, 10.66881903654494686921305557068, 11.56043057792626226741196199905, 12.04198704094714541496910004333