L(s) = 1 | − i·2-s + 2i·3-s − 4-s + 2·6-s − 1.37i·7-s + i·8-s − 9-s − 3.37·11-s − 2i·12-s − 4.74i·13-s − 1.37·14-s + 16-s + 5.37i·17-s + i·18-s + 2·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 1.15i·3-s − 0.5·4-s + 0.816·6-s − 0.518i·7-s + 0.353i·8-s − 0.333·9-s − 1.01·11-s − 0.577i·12-s − 1.31i·13-s − 0.366·14-s + 0.250·16-s + 1.30i·17-s + 0.235i·18-s + 0.458·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7922629210\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7922629210\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 37 | \( 1 + iT \) |
good | 3 | \( 1 - 2iT - 3T^{2} \) |
| 7 | \( 1 + 1.37iT - 7T^{2} \) |
| 11 | \( 1 + 3.37T + 11T^{2} \) |
| 13 | \( 1 + 4.74iT - 13T^{2} \) |
| 17 | \( 1 - 5.37iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 6.74iT - 23T^{2} \) |
| 29 | \( 1 + 8.11T + 29T^{2} \) |
| 31 | \( 1 + 2.62T + 31T^{2} \) |
| 41 | \( 1 - 5.37T + 41T^{2} \) |
| 43 | \( 1 - 7.37iT - 43T^{2} \) |
| 47 | \( 1 - 8.74iT - 47T^{2} \) |
| 53 | \( 1 - 1.37iT - 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 + 5.37T + 61T^{2} \) |
| 67 | \( 1 + 4.74iT - 67T^{2} \) |
| 71 | \( 1 + 6.74T + 71T^{2} \) |
| 73 | \( 1 - 8.74iT - 73T^{2} \) |
| 79 | \( 1 - 4.74T + 79T^{2} \) |
| 83 | \( 1 + 0.744iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 0.116iT - 97T^{2} \) |
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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
−9.656704212083990389238932677146, −9.081025171402562511955456301806, −7.85182212441302617029550061303, −7.57844186878578840532818287865, −5.85287727028271417666382015118, −5.35064578114534100331147534302, −4.37521302056727113435207423538, −3.62684370019102891445776543822, −2.93010456475122841619282505535, −1.43594097789488442093968063949,
0.29192537730937096486011165584, 1.85998818722946274096256526415, 2.73871025669824596314004893493, 4.20480210466807803627250164175, 5.15776692065516030256090098792, 5.91233367601929967764231172926, 6.84839386042119098278054858708, 7.28052702005136397651313086476, 7.959460197475748344955918053625, 8.897248847932190189224620647497