L(s) = 1 | + (0.707 + 0.707i)2-s + (0.707 + 0.707i)3-s + 1.00i·4-s + 1.00i·6-s − i·7-s + (−0.707 + 0.707i)8-s + 1.00i·9-s + (0.707 − 0.707i)11-s + (−0.707 + 0.707i)12-s + (1 + i)13-s + (0.707 − 0.707i)14-s − 1.00·16-s + (−0.707 + 0.707i)18-s + (−1 − i)19-s + (0.707 − 0.707i)21-s + 1.00·22-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s + (0.707 + 0.707i)3-s + 1.00i·4-s + 1.00i·6-s − i·7-s + (−0.707 + 0.707i)8-s + 1.00i·9-s + (0.707 − 0.707i)11-s + (−0.707 + 0.707i)12-s + (1 + i)13-s + (0.707 − 0.707i)14-s − 1.00·16-s + (−0.707 + 0.707i)18-s + (−1 − i)19-s + (0.707 − 0.707i)21-s + 1.00·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.443575148\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.443575148\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (-0.707 + 0.707i)T \) |
good | 5 | \( 1 - iT^{2} \) |
| 13 | \( 1 + (-1 - i)T + iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + (1 + i)T + iT^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (1 - i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 - 1.41T + T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + (1 + i)T + iT^{2} \) |
| 67 | \( 1 + (1 + i)T + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + 2iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + 1.41iT - T^{2} \) |
| 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
−8.873807766994652023005324449491, −8.295088542776908623350237662592, −7.31385970621463403400752210714, −6.72765309507651622351723774079, −6.06961306049964636837109304400, −4.82250243510699597042440576060, −4.45536594150200785989815550951, −3.55325058560849516324945978298, −3.14373104572075453996375183657, −1.66579125784338085115644919254,
1.13991566881758156261962706249, 2.16188520404857814026402643962, 2.72283204690217649658561186276, 3.77604884928185438887531954136, 4.34236286507127342863947020479, 5.66032743066166085025267920487, 6.11423620061491937610954553254, 6.75606638973113922420770446133, 7.88404971215364050732638606452, 8.610146216357638085814195528198