L(s) = 1 | − i·2-s − 4-s − i·7-s + i·8-s + 9-s − 14-s + 16-s − i·18-s − 2i·23-s + i·28-s − i·32-s − 36-s − 2·46-s − 49-s + 56-s + ⋯ |
L(s) = 1 | − i·2-s − 4-s − i·7-s + i·8-s + 9-s − 14-s + 16-s − i·18-s − 2i·23-s + i·28-s − i·32-s − 36-s − 2·46-s − 49-s + 56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.012088152\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.012088152\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + 2iT - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + 2T + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - 2T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 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
−9.765012748437688413368988412335, −8.866905105064671329983701953689, −8.002590247929937488962729954525, −7.16429588126950139333433034383, −6.23411258645905093824127678128, −4.83873280622959609951008350304, −4.32135749502739859126499642656, −3.42991126237455350429414279205, −2.19301970758838871135275432597, −0.954165853816513525998763122907,
1.62674609263186377695371728245, 3.24988279972240346684076149846, 4.27687517674649775287371903480, 5.22797957112767086184052552108, 5.87814879234559160367791851818, 6.81684790072022592688546797161, 7.55305362595740159195052512078, 8.278973365732278437695112623326, 9.273086296306183427017120347383, 9.602136215530039227366546270583