| L(s) = 1 | − 2.04·5-s − 7.79·7-s + 4.09·11-s + 6.69i·13-s − 19.3i·17-s − 1.79i·19-s + 14.1i·23-s − 20.7·25-s − 28.4·29-s + 20.2·31-s + 15.9·35-s + 41.5i·37-s − 4.94i·41-s − 75.1i·43-s + 13.5i·47-s + ⋯ |
| L(s) = 1 | − 0.409·5-s − 1.11·7-s + 0.372·11-s + 0.515i·13-s − 1.13i·17-s − 0.0946i·19-s + 0.614i·23-s − 0.831·25-s − 0.982·29-s + 0.651·31-s + 0.456·35-s + 1.12i·37-s − 0.120i·41-s − 1.74i·43-s + 0.288i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.208637000\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.208637000\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2.04T + 25T^{2} \) |
| 7 | \( 1 + 7.79T + 49T^{2} \) |
| 11 | \( 1 - 4.09T + 121T^{2} \) |
| 13 | \( 1 - 6.69iT - 169T^{2} \) |
| 17 | \( 1 + 19.3iT - 289T^{2} \) |
| 19 | \( 1 + 1.79iT - 361T^{2} \) |
| 23 | \( 1 - 14.1iT - 529T^{2} \) |
| 29 | \( 1 + 28.4T + 841T^{2} \) |
| 31 | \( 1 - 20.2T + 961T^{2} \) |
| 37 | \( 1 - 41.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 4.94iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 75.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 13.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 20.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + 54.8T + 3.48e3T^{2} \) |
| 61 | \( 1 + 89.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 37.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 117. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 48.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 92.6T + 6.24e3T^{2} \) |
| 83 | \( 1 - 158.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 121. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 167.T + 9.40e3T^{2} \) |
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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
−9.046244505219323498508627624329, −8.017544880936939656274082020811, −7.20547113381722045156322792653, −6.62951390770084685222103449595, −5.78664279846017226157758532523, −4.82630575932323262969276918091, −3.83022838588760556018620292141, −3.19491526875142530166884939943, −2.04640390722187018422842514716, −0.58552509270506981506558727903,
0.52234396438325919311684322418, 1.95328847629331828979626171514, 3.17491209708749436428117697743, 3.79435851186618323025925064655, 4.69499869718054972162784887813, 6.04409559667578824956526075914, 6.20119661754942153607753913197, 7.36236117382693131290244229592, 7.962945802397209441844757508799, 8.862928192004373398190284602064
