L(s) = 1 | − 2-s + 4-s + i·7-s − 8-s + (1 − i)11-s − i·14-s + 16-s + (−1 + i)22-s + 2i·23-s − i·25-s + i·28-s + (1 + i)29-s − 32-s + (1 − i)37-s + (−1 + i)43-s + (1 − i)44-s + ⋯ |
L(s) = 1 | − 2-s + 4-s + i·7-s − 8-s + (1 − i)11-s − i·14-s + 16-s + (−1 + i)22-s + 2i·23-s − i·25-s + i·28-s + (1 + i)29-s − 32-s + (1 − i)37-s + (−1 + i)43-s + (1 − i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7071613003\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7071613003\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 + iT^{2} \) |
| 11 | \( 1 + (-1 + i)T - iT^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 - 2iT - T^{2} \) |
| 29 | \( 1 + (-1 - i)T + iT^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-1 + i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (1 - i)T - iT^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-1 + i)T - iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 - iT^{2} \) |
| 67 | \( 1 + (-1 - i)T + iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.982434412954360767712467185201, −9.305990133448545052304525943166, −8.639569538167689318923654389682, −8.004401837670408523660031767230, −6.86910129711472399896744232013, −6.12340794420564508803409601509, −5.34677774913906430353739313483, −3.68289635030729316521850448797, −2.68694585323491520590100941810, −1.36943928948886245491131387362,
1.12320597885965016683311254231, 2.43548912856909220961327135444, 3.81462592280659760118238998911, 4.77468552206478936045536265311, 6.35610132577627102334896989416, 6.80688691232368006684755012941, 7.66253854758341806487239981631, 8.483510857792067554128550925762, 9.374606087729288842602497887666, 10.09448328296181404329669010090