L(s) = 1 | + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (0.241 − 1.83i)5-s + (−0.707 + 0.707i)8-s + (−0.258 − 0.965i)9-s + (0.241 + 1.83i)10-s − 2i·13-s + (0.500 − 0.866i)16-s + (0.965 + 0.258i)17-s + (0.499 + 0.866i)18-s + (−0.707 − 1.70i)20-s + (−2.33 − 0.624i)25-s + (0.517 + 1.93i)26-s + (−0.707 + 0.292i)29-s + (−0.258 + 0.965i)32-s + ⋯ |
L(s) = 1 | + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (0.241 − 1.83i)5-s + (−0.707 + 0.707i)8-s + (−0.258 − 0.965i)9-s + (0.241 + 1.83i)10-s − 2i·13-s + (0.500 − 0.866i)16-s + (0.965 + 0.258i)17-s + (0.499 + 0.866i)18-s + (−0.707 − 1.70i)20-s + (−2.33 − 0.624i)25-s + (0.517 + 1.93i)26-s + (−0.707 + 0.292i)29-s + (−0.258 + 0.965i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7778355097\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7778355097\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 - 0.258i)T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (-0.965 - 0.258i)T \) |
good | 3 | \( 1 + (0.258 + 0.965i)T^{2} \) |
| 5 | \( 1 + (-0.241 + 1.83i)T + (-0.965 - 0.258i)T^{2} \) |
| 11 | \( 1 + (-0.965 + 0.258i)T^{2} \) |
| 13 | \( 1 + 2iT - T^{2} \) |
| 19 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (0.258 - 0.965i)T^{2} \) |
| 29 | \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 + (0.258 + 0.965i)T^{2} \) |
| 37 | \( 1 + (-0.758 - 0.0999i)T + (0.965 + 0.258i)T^{2} \) |
| 41 | \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (-0.607 + 0.465i)T + (0.258 - 0.965i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 73 | \( 1 + (-1.46 - 1.12i)T + (0.258 + 0.965i)T^{2} \) |
| 79 | \( 1 + (0.258 - 0.965i)T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \) |
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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
−8.347952799196216630490519071299, −8.176242912750809353650329190624, −7.35882231727759788232802058228, −6.07815326328273354642102639740, −5.66843160738126186281360472192, −5.05558723972218172657075690660, −3.78637001134685705429494871417, −2.75081087970167207113178591314, −1.31903512894637559127898255580, −0.67888575579658642926529406486,
1.84296255894846897247438665781, 2.38651959579585647361453192512, 3.27575185498820870117200127696, 4.15478408485584330926478866515, 5.58454827928969863583640148768, 6.40990143261304757910065565531, 6.98654739551878334975800643616, 7.55334101167875157536864105107, 8.221232348990862306308820341086, 9.350692435005817593531460174345