L(s) = 1 | + i·2-s − 4-s + i·7-s − i·8-s − 2i·13-s − 14-s + 16-s + 6i·17-s − 8·19-s + 2·26-s − i·28-s + 6·29-s − 4·31-s + i·32-s − 6·34-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.377i·7-s − 0.353i·8-s − 0.554i·13-s − 0.267·14-s + 0.250·16-s + 1.45i·17-s − 1.83·19-s + 0.392·26-s − 0.188i·28-s + 1.11·29-s − 0.718·31-s + 0.176i·32-s − 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−8.394663982777576601807785052690, −7.79856665122842137546509773647, −6.85961694364785670576843052593, −6.08766621867549647746327852455, −5.67703750528507140179191063606, −4.53507047752588653882541464811, −3.95157743770693289972200245937, −2.77879340538732471404902232733, −1.65176570952002498107029789879, 0,
1.34327336253784489312356493318, 2.43461805855537396249455206233, 3.22369915290844804845614088379, 4.43238676642480667306255207838, 4.64674453580542058632471879852, 5.90735893405611094874501195460, 6.68710253522566000263131383101, 7.46099889733317650240924556944, 8.339330707143783256402445841106