L(s) = 1 | − 2i·3-s − i·7-s − 9-s − 3·11-s + 4i·13-s + 2·19-s − 2·21-s − 3i·23-s − 4i·27-s − 9·29-s − 8·31-s + 6i·33-s + 5i·37-s + 8·39-s − 6·41-s + ⋯ |
L(s) = 1 | − 1.15i·3-s − 0.377i·7-s − 0.333·9-s − 0.904·11-s + 1.10i·13-s + 0.458·19-s − 0.436·21-s − 0.625i·23-s − 0.769i·27-s − 1.67·29-s − 1.43·31-s + 1.04i·33-s + 0.821i·37-s + 1.28·39-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 3iT - 23T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 5iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 11iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 5iT - 67T^{2} \) |
| 71 | \( 1 + 15T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 7T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−8.058823179046715229263175723914, −7.46181941086381589019593464721, −6.91959438946699318308594335924, −6.18628571674854184302972098868, −5.25885429901591757925485511255, −4.35608370375259282853289871296, −3.31419399824660808740319286938, −2.16878031623773248339956403444, −1.44325447619435435747505969879, 0,
1.84758571978563807409937852785, 3.10430847783344829519098190044, 3.65526627989818650491721248992, 4.70804525865876651982136532390, 5.49416752198638095866245041426, 5.79250126945184046131368402227, 7.41844493321202473710536866686, 7.60791193984134380716649530272, 8.913383109175226540092754157803