L(s) = 1 | + (0.5 − 0.866i)5-s + 7-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)11-s + 13-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)23-s + (−0.499 − 0.866i)25-s + (0.5 − 0.866i)35-s + (−0.5 + 0.866i)37-s − 41-s + (0.499 + 0.866i)45-s + (0.5 − 0.866i)47-s + 49-s + (−0.5 − 0.866i)53-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)5-s + 7-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)11-s + 13-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)23-s + (−0.499 − 0.866i)25-s + (0.5 − 0.866i)35-s + (−0.5 + 0.866i)37-s − 41-s + (0.499 + 0.866i)45-s + (0.5 − 0.866i)47-s + 49-s + (−0.5 − 0.866i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.208792625\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.208792625\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 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
−10.16846576731008600779091813970, −8.646081975701094140525373473946, −8.580396719590641342777347651911, −7.84446045348975225843703643558, −6.44518845529571995301613283290, −5.50814084261104868298984074770, −5.03480633452826466468459673362, −3.91875780038000633101148288845, −2.47700450651409867859337629618, −1.35704853684034896538850773240,
1.66369959513530134044079342598, 2.79541741287018713697277611429, 3.86139378875084893608000809995, 5.04240510705274842988192381942, 5.89115990679894339211422985552, 6.78835960436435921430175173794, 7.50577208572087090095483399110, 8.537488976109723831314488477146, 9.255704456255529216337537013479, 10.13391253681635970545600693948