L(s) = 1 | + (−0.382 − 0.923i)2-s + (−0.707 + 0.707i)3-s + (−0.707 + 0.707i)4-s + (0.923 − 0.382i)5-s + (0.923 + 0.382i)6-s + (0.923 + 0.382i)8-s − 1.00i·9-s + (−0.707 − 0.707i)10-s + 0.765·11-s − i·12-s + (−0.707 − 0.707i)13-s + (−0.382 + 0.923i)15-s − i·16-s + (−0.923 + 0.382i)18-s + (−0.382 + 0.923i)20-s + ⋯ |
L(s) = 1 | + (−0.382 − 0.923i)2-s + (−0.707 + 0.707i)3-s + (−0.707 + 0.707i)4-s + (0.923 − 0.382i)5-s + (0.923 + 0.382i)6-s + (0.923 + 0.382i)8-s − 1.00i·9-s + (−0.707 − 0.707i)10-s + 0.765·11-s − i·12-s + (−0.707 − 0.707i)13-s + (−0.382 + 0.923i)15-s − i·16-s + (−0.923 + 0.382i)18-s + (−0.382 + 0.923i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8034883382\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8034883382\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.382 + 0.923i)T \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (-0.923 + 0.382i)T \) |
| 13 | \( 1 + (0.707 + 0.707i)T \) |
good | 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 - 0.765T + T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 - 1.84T + T^{2} \) |
| 43 | \( 1 + (1 - i)T - iT^{2} \) |
| 47 | \( 1 + (-1.30 + 1.30i)T - iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - 1.84iT - T^{2} \) |
| 61 | \( 1 + 1.41iT - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + 0.765iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - 1.41T + T^{2} \) |
| 83 | \( 1 + (-0.541 + 0.541i)T - iT^{2} \) |
| 89 | \( 1 - 0.765iT - T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.549610932075097916263751937172, −9.174202748822408133184977206499, −8.247750261992013122547574318756, −7.08214084386833621888387650479, −6.02821804348682949487856231211, −5.19705984555585452789739430562, −4.47119152313669216391937433004, −3.47981297561342951219708081132, −2.32081276718488643788105210728, −0.959729820655317169865916482465,
1.28207386368279233686040014983, 2.36732724401881012088599707655, 4.21844466512772928948418280886, 5.17257152590291313623052508560, 5.93910138875777311344453203469, 6.57507599549097401905487457339, 7.14603080215316213870064565859, 7.915834217001041008883125745270, 9.072140980926536982262349448714, 9.530875036643354623468239954209