L(s) = 1 | + (−1.64 − 1.12i)3-s + (−0.0747 + 0.997i)5-s + (0.294 + 0.955i)7-s + (1.08 + 2.77i)9-s + (1.24 − 1.55i)15-s + (0.587 − 1.90i)21-s + (0.825 + 0.766i)23-s + (−0.988 − 0.149i)25-s + (0.877 − 3.84i)27-s + (0.326 + 1.42i)29-s + (−0.974 + 0.222i)35-s + (0.658 − 0.317i)41-s + (−1.67 − 0.807i)43-s + (−2.84 + 0.877i)45-s + (−0.858 + 0.129i)47-s + ⋯ |
L(s) = 1 | + (−1.64 − 1.12i)3-s + (−0.0747 + 0.997i)5-s + (0.294 + 0.955i)7-s + (1.08 + 2.77i)9-s + (1.24 − 1.55i)15-s + (0.587 − 1.90i)21-s + (0.825 + 0.766i)23-s + (−0.988 − 0.149i)25-s + (0.877 − 3.84i)27-s + (0.326 + 1.42i)29-s + (−0.974 + 0.222i)35-s + (0.658 − 0.317i)41-s + (−1.67 − 0.807i)43-s + (−2.84 + 0.877i)45-s + (−0.858 + 0.129i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5241405220\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5241405220\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.0747 - 0.997i)T \) |
| 7 | \( 1 + (-0.294 - 0.955i)T \) |
good | 3 | \( 1 + (1.64 + 1.12i)T + (0.365 + 0.930i)T^{2} \) |
| 11 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 13 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 17 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.825 - 0.766i)T + (0.0747 + 0.997i)T^{2} \) |
| 29 | \( 1 + (-0.326 - 1.42i)T + (-0.900 + 0.433i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 41 | \( 1 + (-0.658 + 0.317i)T + (0.623 - 0.781i)T^{2} \) |
| 43 | \( 1 + (1.67 + 0.807i)T + (0.623 + 0.781i)T^{2} \) |
| 47 | \( 1 + (0.858 - 0.129i)T + (0.955 - 0.294i)T^{2} \) |
| 53 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 59 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 61 | \( 1 + (1.57 + 0.487i)T + (0.826 + 0.563i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 73 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.367 - 0.460i)T + (-0.222 - 0.974i)T^{2} \) |
| 89 | \( 1 + (0.535 + 1.36i)T + (-0.733 + 0.680i)T^{2} \) |
| 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
−8.647867694712537076453870975439, −7.83648660255439544865387622927, −7.13347516265540453652566476550, −6.68012493099282379851851220754, −5.94603131810895039671006139279, −5.35060482645361579536175964203, −4.71753589840573262156111588504, −3.25317729848212366582835938725, −2.21934295675080814997973944251, −1.39376095684212775472994112153,
0.41153599416052614020306215351, 1.37460365566642627860684542950, 3.35567923168536996606989024828, 4.28561922367763994054996416934, 4.64373393126515580893451219301, 5.22428045461897421730421520670, 6.14216644705178014284452510836, 6.66961535047202700466383022750, 7.68055461915159470308243482856, 8.554715546265628577360048418259