| L(s) = 1 | − i·2-s − 4-s + 3.90·5-s + (1.49 + 2.18i)7-s + i·8-s − 3.90i·10-s − i·11-s + 6.17i·13-s + (2.18 − 1.49i)14-s + 16-s + 0.462·17-s + 1.61i·19-s − 3.90·20-s − 22-s + 10.2·25-s + 6.17·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 1.74·5-s + (0.565 + 0.824i)7-s + 0.353i·8-s − 1.23i·10-s − 0.301i·11-s + 1.71i·13-s + (0.583 − 0.399i)14-s + 0.250·16-s + 0.112·17-s + 0.370i·19-s − 0.872·20-s − 0.213·22-s + 2.04·25-s + 1.21·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0148i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.283352608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.283352608\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (-1.49 - 2.18i)T \) |
| 11 | \( 1 + iT \) |
| good | 5 | \( 1 - 3.90T + 5T^{2} \) |
| 13 | \( 1 - 6.17iT - 13T^{2} \) |
| 17 | \( 1 - 0.462T + 17T^{2} \) |
| 19 | \( 1 - 1.61iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 7.80iT - 29T^{2} \) |
| 31 | \( 1 + 5.98iT - 31T^{2} \) |
| 37 | \( 1 + 9.05T + 37T^{2} \) |
| 41 | \( 1 - 0.191T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 - 5.05T + 47T^{2} \) |
| 53 | \( 1 - 6.04iT - 53T^{2} \) |
| 59 | \( 1 - 4.54T + 59T^{2} \) |
| 61 | \( 1 + 2.55iT - 61T^{2} \) |
| 67 | \( 1 + 5.80T + 67T^{2} \) |
| 71 | \( 1 + 8.48iT - 71T^{2} \) |
| 73 | \( 1 + 10.8iT - 73T^{2} \) |
| 79 | \( 1 - 16.2T + 79T^{2} \) |
| 83 | \( 1 - 9.86T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + 11.0iT - 97T^{2} \) |
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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
−9.396663503867837363181486225457, −9.140432598353876499702389065154, −8.351906932807675370886779156259, −6.94899861202569651111992739846, −6.10133328739375392461265990341, −5.36673292808384401472538106983, −4.60277141034025590243414017985, −3.21857185616279086236341376007, −2.00424207055247319798801922066, −1.67462866062675116638573537678,
0.994323583622214011312277991141, 2.26399579118988848309903113561, 3.55326277895259083078918983896, 4.99633737399651832079745662257, 5.33032424077675088764521741230, 6.30324788627602512439921975060, 7.02850563511414648575342802011, 7.949527854814425151612814895604, 8.677086050374408274182893999226, 9.665653151229143903993061390513
