L(s) = 1 | + (0.541 + 0.541i)2-s − 0.414i·4-s + (0.382 + 0.923i)5-s + (0.765 − 0.765i)8-s + (−0.292 + 0.707i)10-s + 0.765i·11-s + (−0.707 − 0.707i)13-s + 0.414·16-s + (0.382 − 0.158i)20-s + (−0.414 + 0.414i)22-s + (−0.707 + 0.707i)25-s − 0.765i·26-s + (−0.541 − 0.541i)32-s + (1 + 0.414i)40-s + 1.84i·41-s + ⋯ |
L(s) = 1 | + (0.541 + 0.541i)2-s − 0.414i·4-s + (0.382 + 0.923i)5-s + (0.765 − 0.765i)8-s + (−0.292 + 0.707i)10-s + 0.765i·11-s + (−0.707 − 0.707i)13-s + 0.414·16-s + (0.382 − 0.158i)20-s + (−0.414 + 0.414i)22-s + (−0.707 + 0.707i)25-s − 0.765i·26-s + (−0.541 − 0.541i)32-s + (1 + 0.414i)40-s + 1.84i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.248432355\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.248432355\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.382 - 0.923i)T \) |
| 13 | \( 1 + (0.707 + 0.707i)T \) |
good | 2 | \( 1 + (-0.541 - 0.541i)T + iT^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 - 0.765iT - T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 - 1.84iT - T^{2} \) |
| 43 | \( 1 + (1 + i)T + iT^{2} \) |
| 47 | \( 1 + (1.30 + 1.30i)T + iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + 1.84T + T^{2} \) |
| 61 | \( 1 - 1.41T + T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + 1.84iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - 1.41iT - T^{2} \) |
| 83 | \( 1 + (1.30 - 1.30i)T - iT^{2} \) |
| 89 | \( 1 - 0.765T + T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−10.83491246202075359656303133814, −10.03313393293818078659110995364, −9.644704973625450081759550777416, −8.073982273129616687472969930525, −7.10505576036534290204399991449, −6.54310226973302010054823846722, −5.50623439071039200262231745347, −4.72050206365518095967900287044, −3.36298025464857859519113393950, −1.97555697114039737323873690728,
1.77830629201225741388919556009, 3.04133990223048461664853622844, 4.26182268001566047566648137340, 4.99551464274868561556390286452, 6.06539481906566001520305560184, 7.36271585030668874097225486716, 8.312036772387671408730363858142, 9.043279315929973620732128078036, 9.987524839695992137949128078262, 11.10537437956690019417148919381