| L(s) = 1 | + 4.29i·5-s − 2.75i·7-s − 13.7·11-s + 14.5i·13-s − 22.8·17-s − 16.0·19-s − 17.1i·23-s + 6.57·25-s + 21.8i·29-s − 38.6i·31-s + 11.8·35-s − 66.4i·37-s + 23.2·41-s + 47.9·43-s + 14.8i·47-s + ⋯ |
| L(s) = 1 | + 0.858i·5-s − 0.393i·7-s − 1.25·11-s + 1.11i·13-s − 1.34·17-s − 0.844·19-s − 0.743i·23-s + 0.262·25-s + 0.754i·29-s − 1.24i·31-s + 0.338·35-s − 1.79i·37-s + 0.566·41-s + 1.11·43-s + 0.315i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.121180447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.121180447\) |
| \(L(2)\) |
|
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 | \( 1 - 4.29iT - 25T^{2} \) |
| 7 | \( 1 + 2.75iT - 49T^{2} \) |
| 11 | \( 1 + 13.7T + 121T^{2} \) |
| 13 | \( 1 - 14.5iT - 169T^{2} \) |
| 17 | \( 1 + 22.8T + 289T^{2} \) |
| 19 | \( 1 + 16.0T + 361T^{2} \) |
| 23 | \( 1 + 17.1iT - 529T^{2} \) |
| 29 | \( 1 - 21.8iT - 841T^{2} \) |
| 31 | \( 1 + 38.6iT - 961T^{2} \) |
| 37 | \( 1 + 66.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 23.2T + 1.68e3T^{2} \) |
| 43 | \( 1 - 47.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 14.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 65.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 65.8T + 3.48e3T^{2} \) |
| 61 | \( 1 + 40.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 74.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + 122. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 144.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 128. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 22.0T + 6.88e3T^{2} \) |
| 89 | \( 1 + 122.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 88.3T + 9.40e3T^{2} \) |
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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
−8.811687470831390253792885448297, −7.83364424953780095437930366084, −7.11735857212505051711501152319, −6.54815452206563498946773977006, −5.69615998097253527178309555067, −4.52971442064771512217501320605, −3.99264208470539228457921560724, −2.63135997253282899583759804919, −2.15571437042288559105393110833, −0.34943590090934267494607594539,
0.797773314815004366025184356609, 2.17564437960394385498553399956, 2.98149981119600202269919023448, 4.24026988304392426677794688258, 5.05394884741171457811465555155, 5.58063205713414596002231856146, 6.55335403413662399689393922311, 7.51305371522198091446946363262, 8.437457426520524864122010832266, 8.605677356068915123229731559614
