| L(s) = 1 | + i·2-s + 3-s − 4-s + i·5-s + i·6-s − 1.30·7-s − i·8-s + 9-s − 10-s + 0.690·11-s − 12-s − 2.69i·13-s − 1.30i·14-s + i·15-s + 16-s + 6.90i·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.5·4-s + 0.447i·5-s + 0.408i·6-s − 0.495·7-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s + 0.208·11-s − 0.288·12-s − 0.746i·13-s − 0.350i·14-s + 0.258i·15-s + 0.250·16-s + 1.67i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.675 - 0.737i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.519892147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.519892147\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - iT \) |
| 37 | \( 1 + (-4.10 - 4.48i)T \) |
| good | 7 | \( 1 + 1.30T + 7T^{2} \) |
| 11 | \( 1 - 0.690T + 11T^{2} \) |
| 13 | \( 1 + 2.69iT - 13T^{2} \) |
| 17 | \( 1 - 6.90iT - 17T^{2} \) |
| 19 | \( 1 - 2.69iT - 19T^{2} \) |
| 23 | \( 1 - 8.90iT - 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 - 9.59iT - 31T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 8.21iT - 43T^{2} \) |
| 47 | \( 1 + 4.61T + 47T^{2} \) |
| 53 | \( 1 - 0.690T + 53T^{2} \) |
| 59 | \( 1 + 6.21iT - 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 + 1.23T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 1.59iT - 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 5.92iT - 89T^{2} \) |
| 97 | \( 1 - 3.38iT - 97T^{2} \) |
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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
−10.04588279933314269737514205377, −9.248366886282480204063696603828, −8.287788383643369800637694671990, −7.78808809279929974318178123569, −6.79340467680432250309696574288, −6.08961524103645144034643711267, −5.16724188425490238388222900796, −3.77142648111712192071763491712, −3.27379169299567488510848209138, −1.63653399655713109092775493772,
0.63660432305458623611981268130, 2.21650449288896423544958415927, 3.01963307213901217309815872341, 4.25342075985463205402851253593, 4.83304020450057745203569416331, 6.20363989591515745785422576131, 7.11846821706763686852109765434, 8.093928667592776602905139561298, 9.077454236990862449874736293072, 9.385312127071128340443376349399
