L(s) = 1 | + (0.634 + 0.773i)2-s + (−0.195 + 0.980i)4-s + (−0.980 + 0.195i)5-s + (0.674 + 1.62i)7-s + (−0.881 + 0.471i)8-s + (0.382 − 0.923i)9-s + (−0.773 − 0.634i)10-s + (0.555 − 0.831i)11-s + (1.95 + 0.388i)13-s + (−0.831 + 1.55i)14-s + (−0.923 − 0.382i)16-s + (0.410 + 0.410i)17-s + (0.956 − 0.290i)18-s − i·20-s + (0.995 − 0.0980i)22-s + ⋯ |
L(s) = 1 | + (0.634 + 0.773i)2-s + (−0.195 + 0.980i)4-s + (−0.980 + 0.195i)5-s + (0.674 + 1.62i)7-s + (−0.881 + 0.471i)8-s + (0.382 − 0.923i)9-s + (−0.773 − 0.634i)10-s + (0.555 − 0.831i)11-s + (1.95 + 0.388i)13-s + (−0.831 + 1.55i)14-s + (−0.923 − 0.382i)16-s + (0.410 + 0.410i)17-s + (0.956 − 0.290i)18-s − i·20-s + (0.995 − 0.0980i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.770922642\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.770922642\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.634 - 0.773i)T \) |
| 5 | \( 1 + (0.980 - 0.195i)T \) |
| 11 | \( 1 + (-0.555 + 0.831i)T \) |
good | 3 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 7 | \( 1 + (-0.674 - 1.62i)T + (-0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 + (-1.95 - 0.388i)T + (0.923 + 0.382i)T^{2} \) |
| 17 | \( 1 + (-0.410 - 0.410i)T + iT^{2} \) |
| 19 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 23 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 29 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 31 | \( 1 - 1.11iT - T^{2} \) |
| 37 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 41 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 43 | \( 1 + (1.28 + 0.858i)T + (0.382 + 0.923i)T^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 59 | \( 1 + (1.92 - 0.382i)T + (0.923 - 0.382i)T^{2} \) |
| 61 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 67 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 71 | \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 73 | \( 1 + (0.485 - 1.17i)T + (-0.707 - 0.707i)T^{2} \) |
| 79 | \( 1 + iT^{2} \) |
| 83 | \( 1 + (-0.388 + 1.95i)T + (-0.923 - 0.382i)T^{2} \) |
| 89 | \( 1 + (-0.360 + 0.149i)T + (0.707 - 0.707i)T^{2} \) |
| 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
−8.713460135750620622510461918324, −8.412396649608657841674787947692, −7.51192283531433781878514679255, −6.40591425056581279047087193506, −6.22111810835977859517748712176, −5.33956399205709931894337769624, −4.37941681369750435134725379185, −3.54383096261129839355938384099, −3.14515885182859583354605494550, −1.50540542592213876847745986226,
1.02037887018436987473391492293, 1.67911965593981375059883260335, 3.25561432071524470367780052008, 3.95428312719627803727604996526, 4.44305118500238693737433506986, 5.05950391511809458476401139245, 6.24273949188408881266490586470, 7.09189325049047379655082465690, 7.78915718636991014319035282544, 8.352293907976534771683988780706