L(s) = 1 | + (1 + i)2-s + i·4-s + (0.707 − 0.707i)7-s − 1.41i·11-s + 1.41·14-s + 16-s + (1.41 − 1.41i)22-s + (−1 + i)23-s + (0.707 + 0.707i)28-s − 1.41·29-s + (1 + i)32-s + (−1.41 + 1.41i)37-s + (1.41 + 1.41i)43-s + 1.41·44-s − 2·46-s + ⋯ |
L(s) = 1 | + (1 + i)2-s + i·4-s + (0.707 − 0.707i)7-s − 1.41i·11-s + 1.41·14-s + 16-s + (1.41 − 1.41i)22-s + (−1 + i)23-s + (0.707 + 0.707i)28-s − 1.41·29-s + (1 + i)32-s + (−1.41 + 1.41i)37-s + (1.41 + 1.41i)43-s + 1.41·44-s − 2·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.003804582\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.003804582\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 2 | \( 1 + (-1 - i)T + iT^{2} \) |
| 11 | \( 1 + 1.41iT - T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (1 - i)T - iT^{2} \) |
| 29 | \( 1 + 1.41T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (1 - i)T - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 - 1.41iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + 2iT - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 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
−9.678437011252515424707158164037, −8.577980179260584985819491918164, −7.81828459066011525662042683404, −7.30946607731552893033521420475, −6.24250962183797518004255215219, −5.70434431063958509331185228818, −4.85393285231837693529102005690, −3.97280998433675398358814642686, −3.25025122499728783537351182356, −1.42859601942606664496582095775,
1.92967122841025995921833626643, 2.24318965791129393154419731591, 3.64624729068041528649182521423, 4.40736471351205059243657853202, 5.16386153263934164796902579443, 5.86943550999571355956861441954, 7.14548019243166104953283662636, 7.908802725292050503752832779602, 8.868116735997220002653907211757, 9.734607599617131855971131472693