| L(s) = 1 | + 2.82·3-s + 4i·5-s + 11.3i·7-s − 0.999·9-s + 14.1·11-s − 20i·13-s + 11.3i·15-s − 10·17-s + 14.1·19-s + 32.0i·21-s − 11.3i·23-s + 9·25-s − 28.2·27-s − 20i·29-s + 40.0·33-s + ⋯ |
| L(s) = 1 | + 0.942·3-s + 0.800i·5-s + 1.61i·7-s − 0.111·9-s + 1.28·11-s − 1.53i·13-s + 0.754i·15-s − 0.588·17-s + 0.744·19-s + 1.52i·21-s − 0.491i·23-s + 0.359·25-s − 1.04·27-s − 0.689i·29-s + 1.21·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.69989 + 0.704121i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.69989 + 0.704121i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 2.82T + 9T^{2} \) |
| 5 | \( 1 - 4iT - 25T^{2} \) |
| 7 | \( 1 - 11.3iT - 49T^{2} \) |
| 11 | \( 1 - 14.1T + 121T^{2} \) |
| 13 | \( 1 + 20iT - 169T^{2} \) |
| 17 | \( 1 + 10T + 289T^{2} \) |
| 19 | \( 1 - 14.1T + 361T^{2} \) |
| 23 | \( 1 + 11.3iT - 529T^{2} \) |
| 29 | \( 1 + 20iT - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 20iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 30T + 1.68e3T^{2} \) |
| 43 | \( 1 - 2.82T + 1.84e3T^{2} \) |
| 47 | \( 1 + 67.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 60iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 42.4T + 3.48e3T^{2} \) |
| 61 | \( 1 - 28iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 82.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 56.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 10T + 5.32e3T^{2} \) |
| 79 | \( 1 - 113. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 25.4T + 6.88e3T^{2} \) |
| 89 | \( 1 - 22T + 7.92e3T^{2} \) |
| 97 | \( 1 - 150T + 9.40e3T^{2} \) |
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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
−13.35990853019363253347221366306, −12.16128516751014286824096339211, −11.31103822379610936421014180384, −9.896461189287653538246542980095, −8.880111414089187812477907320460, −8.162882255535140642868467621067, −6.64576054364390513598877879113, −5.47452562483692818414742060445, −3.37834201385038532727259609759, −2.43856292311337517798757803464,
1.42153884939332256783519220759, 3.63831957008523075211197487405, 4.56751666166245411176518987689, 6.61760542648625126507238439830, 7.62505809827393701280915425657, 8.974084526228526313735321609900, 9.407267698132585691184610064109, 10.95865272397958724854320003611, 11.96264894314212811788843553039, 13.31881621428017755010590356690
