| L(s) = 1 | + 4.78i·5-s + 11.8·7-s + 3.31i·11-s − 11.8·13-s + 32.9i·17-s + 20.0·19-s − 31.8i·23-s + 2.11·25-s + 46.9i·29-s + 27.7·31-s + 56.5i·35-s − 63.8·37-s + 4.45i·41-s − 20.4·43-s − 72.2i·47-s + ⋯ |
| L(s) = 1 | + 0.956i·5-s + 1.68·7-s + 0.301i·11-s − 0.913·13-s + 1.93i·17-s + 1.05·19-s − 1.38i·23-s + 0.0844·25-s + 1.61i·29-s + 0.894·31-s + 1.61i·35-s − 1.72·37-s + 0.108i·41-s − 0.475·43-s − 1.53i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1188 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1188 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.181152072\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.181152072\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - 3.31iT \) |
| good | 5 | \( 1 - 4.78iT - 25T^{2} \) |
| 7 | \( 1 - 11.8T + 49T^{2} \) |
| 13 | \( 1 + 11.8T + 169T^{2} \) |
| 17 | \( 1 - 32.9iT - 289T^{2} \) |
| 19 | \( 1 - 20.0T + 361T^{2} \) |
| 23 | \( 1 + 31.8iT - 529T^{2} \) |
| 29 | \( 1 - 46.9iT - 841T^{2} \) |
| 31 | \( 1 - 27.7T + 961T^{2} \) |
| 37 | \( 1 + 63.8T + 1.36e3T^{2} \) |
| 41 | \( 1 - 4.45iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 20.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 72.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 6.97iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 10.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 25.0T + 3.72e3T^{2} \) |
| 67 | \( 1 - 48.1T + 4.48e3T^{2} \) |
| 71 | \( 1 - 12.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 41.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + 113.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 121. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 130. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 98.6T + 9.40e3T^{2} \) |
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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
−10.09917017909731018451994456961, −8.682016142193287071334344054110, −8.211176411246814315926042690531, −7.24623996279374493258788055605, −6.65498362557540457743656080280, −5.37835061859915932790383224650, −4.72870653335324649729698589682, −3.60710650021332746975840878834, −2.39569915601341632861281478357, −1.44107124035687800555527166508,
0.67989822400422896359428325785, 1.72461094808579109956644274588, 2.99318809321559214542346592905, 4.53500377773030232197246522120, 4.97964586018812520272094236524, 5.62836854128819792439597335512, 7.22452155647545959901004134924, 7.71288916729872703306167096541, 8.518188625093100750695130381539, 9.345658650083635385646056305433
