| L(s) = 1 | − i·2-s + 4-s + (−1 + 2i)5-s − 2i·7-s − 3i·8-s + 3·9-s + (2 + i)10-s − 4·11-s + 2i·13-s − 2·14-s − 16-s − 4i·17-s − 3i·18-s + (−1 + 2i)20-s + 4i·22-s − 6i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.5·4-s + (−0.447 + 0.894i)5-s − 0.755i·7-s − 1.06i·8-s + 9-s + (0.632 + 0.316i)10-s − 1.20·11-s + 0.554i·13-s − 0.534·14-s − 0.250·16-s − 0.970i·17-s − 0.707i·18-s + (−0.223 + 0.447i)20-s + 0.852i·22-s − 1.25i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.644165658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.644165658\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1 - 2i)T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + iT - 2T^{2} \) |
| 3 | \( 1 - 3T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 18iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−9.363847935394736232068273008489, −7.950771934631117162400433233461, −7.25056949826501419901367460831, −6.98760151886358534286511560451, −5.93921632520865009025170395212, −4.52349203961254367940021236201, −3.90598276121166886821548053738, −2.84280678188279091484827638574, −2.13120344577616346474351662927, −0.60121922839341753337741176985,
1.43732905366755316359508533950, 2.53916964481262581412084284365, 3.78153311591346673780064539915, 4.94231025974138457774315043763, 5.50611793637179759475933898353, 6.23925995807841324328217601146, 7.44132017131287917434802625896, 7.80498990938258953444581486176, 8.468562224646369129477604645478, 9.362531116971785895102511547506
