L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s − 2i·7-s − i·8-s − 9-s − 3·11-s + i·12-s + 6i·13-s + 2·14-s + 16-s − 2i·17-s − i·18-s + 19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.755i·7-s − 0.353i·8-s − 0.333·9-s − 0.904·11-s + 0.288i·12-s + 1.66i·13-s + 0.534·14-s + 0.250·16-s − 0.485i·17-s − 0.235i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.399442485\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.399442485\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 23 | \( 1 - iT - 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 + 9iT - 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 + 13T + 61T^{2} \) |
| 67 | \( 1 + 7iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 9iT - 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 - iT - 83T^{2} \) |
| 89 | \( 1 + 5T + 89T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
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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
−8.462279199523224918592703960594, −7.921242446631984372756002799708, −6.98396662738868629969933946053, −6.79702286144541205171218715456, −5.79098249384282533703610905507, −4.85462015337871508270560184677, −4.21588923682454297543402975749, −3.05589714247615099722976532279, −1.89372479066211228136237285980, −0.57507951696368228021810819184,
0.931711828038443377798866922169, 2.61166241734859765591218639953, 2.87140058729412711720948942795, 4.01830198106976228824132513125, 4.97362263765330483404157859474, 5.53247227159167075779657588176, 6.28027786695121415592994528889, 7.69332819003604671499366066574, 8.227442610722895579362172838116, 8.876953180679145020259005279157