| L(s) = 1 | + (−4.30 − 2.54i)5-s + 3.84·7-s + 6.19i·11-s + 16.1i·13-s − 5.20i·17-s − 36.2i·19-s − 22.0·23-s + (12.0 + 21.9i)25-s + 20.0·29-s + 26.4i·31-s + (−16.5 − 9.80i)35-s − 69.3i·37-s − 11.6·41-s − 25.8·43-s + 66.1·47-s + ⋯ |
| L(s) = 1 | + (−0.860 − 0.509i)5-s + 0.549·7-s + 0.562i·11-s + 1.23i·13-s − 0.306i·17-s − 1.90i·19-s − 0.958·23-s + (0.480 + 0.876i)25-s + 0.690·29-s + 0.852i·31-s + (−0.473 − 0.280i)35-s − 1.87i·37-s − 0.283·41-s − 0.601·43-s + 1.40·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.248i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3254234722\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3254234722\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (4.30 + 2.54i)T \) |
| good | 7 | \( 1 - 3.84T + 49T^{2} \) |
| 11 | \( 1 - 6.19iT - 121T^{2} \) |
| 13 | \( 1 - 16.1iT - 169T^{2} \) |
| 17 | \( 1 + 5.20iT - 289T^{2} \) |
| 19 | \( 1 + 36.2iT - 361T^{2} \) |
| 23 | \( 1 + 22.0T + 529T^{2} \) |
| 29 | \( 1 - 20.0T + 841T^{2} \) |
| 31 | \( 1 - 26.4iT - 961T^{2} \) |
| 37 | \( 1 + 69.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 11.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + 25.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 66.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 39.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 27.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 54.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 107.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 70.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 37.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 97.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 126.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 133.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 6.40iT - 9.40e3T^{2} \) |
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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
−8.931352966432017460231408575793, −8.264249239118650716281835993537, −7.23389024247946283501130372142, −6.85358549954763450828013568511, −5.44771376007139552371399680875, −4.55978500244973033170537970381, −4.12236175984586556435083313802, −2.70102046734719850803070448400, −1.50819767116874850376773376154, −0.094338465914286492347373006127,
1.37773351857121842775779198331, 2.85809838144071771129909584143, 3.66138002203070212354855218785, 4.54902009050835202705308712010, 5.72437901826644188677762617013, 6.33149799651509308179484609382, 7.61872626982078523609823573075, 8.028955494374863139854073476380, 8.564113886556151354969510351931, 10.03071385767911404173919971582
