| L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s + 4i·7-s + i·8-s − 9-s + i·12-s − 2i·13-s + 4·14-s + 16-s − 2i·17-s + i·18-s + 19-s + 4·21-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s + 1.51i·7-s + 0.353i·8-s − 0.333·9-s + 0.288i·12-s − 0.554i·13-s + 1.06·14-s + 0.250·16-s − 0.485i·17-s + 0.235i·18-s + 0.229·19-s + 0.872·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7899082115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7899082115\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−9.069886398799138107344701630448, −8.234460279125897784171915564557, −7.59884664048498673843922003330, −6.57608615012675359818066504997, −5.58272876096380122791336356599, −5.31576124936702369664580313342, −4.02043861735270127049854934749, −2.91580271732660290936732476790, −2.38323186119880101096785166899, −1.29904763236614849315474992908,
0.25887487383778662700780252316, 1.72788069589953960569378938951, 3.39778997460266987656486339706, 3.99840375652004299268911124452, 4.67305116665948802341889046723, 5.58723367337564041232111332853, 6.42411380058286349477381271832, 7.24773382618718142579432258969, 7.70418375914744222843625633973, 8.626054588104362840655806160689
