L(s) = 1 | − 2i·3-s + 7-s − 9-s + 4i·13-s + 6·17-s + 2i·19-s − 2i·21-s + 5·25-s − 4i·27-s + 6i·29-s + 4·31-s + 2i·37-s + 8·39-s − 6·41-s − 8i·43-s + ⋯ |
L(s) = 1 | − 1.15i·3-s + 0.377·7-s − 0.333·9-s + 1.10i·13-s + 1.45·17-s + 0.458i·19-s − 0.436i·21-s + 25-s − 0.769i·27-s + 1.11i·29-s + 0.718·31-s + 0.328i·37-s + 1.28·39-s − 0.937·41-s − 1.21i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.988236859\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.988236859\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 5 | \( 1 - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 + 8iT - 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.969894990703285873460521721506, −8.319348796562183905995251752788, −7.42972218565900774766647327548, −6.98830406054245652288390218989, −6.12207400655819751030723132536, −5.22322005488711877978514198455, −4.20828599827424517765801743325, −3.05346149573610183813753309475, −1.85911327474741436580407073794, −1.07240091572505906501474742470,
1.00373239285767889559039585838, 2.70743521323646971641217695588, 3.53830435246007457620500860630, 4.47867284023329427029777746450, 5.20561269830225679526579033922, 5.89274378722533440007963278070, 7.10860822769963937535741884241, 7.942853529295811248914099066303, 8.629298718029248324586048066036, 9.582244117207813306484081522867