| L(s) = 1 | + (−0.463 + 2.18i)5-s + 3.14i·7-s − 2.42i·11-s + 3.12·13-s − 7.15i·17-s − 2.34i·19-s − 1.28i·23-s + (−4.57 − 2.02i)25-s − 4.21i·29-s − 3.05·31-s + (−6.89 − 1.45i)35-s − 1.37·37-s + 11.0·41-s − 9.70·43-s − 0.627i·47-s + ⋯ |
| L(s) = 1 | + (−0.207 + 0.978i)5-s + 1.19i·7-s − 0.730i·11-s + 0.866·13-s − 1.73i·17-s − 0.538i·19-s − 0.268i·23-s + (−0.914 − 0.405i)25-s − 0.782i·29-s − 0.549·31-s + (−1.16 − 0.246i)35-s − 0.225·37-s + 1.72·41-s − 1.47·43-s − 0.0915i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.498 + 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.498 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.245203381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.245203381\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.463 - 2.18i)T \) |
| good | 7 | \( 1 - 3.14iT - 7T^{2} \) |
| 11 | \( 1 + 2.42iT - 11T^{2} \) |
| 13 | \( 1 - 3.12T + 13T^{2} \) |
| 17 | \( 1 + 7.15iT - 17T^{2} \) |
| 19 | \( 1 + 2.34iT - 19T^{2} \) |
| 23 | \( 1 + 1.28iT - 23T^{2} \) |
| 29 | \( 1 + 4.21iT - 29T^{2} \) |
| 31 | \( 1 + 3.05T + 31T^{2} \) |
| 37 | \( 1 + 1.37T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 9.70T + 43T^{2} \) |
| 47 | \( 1 + 0.627iT - 47T^{2} \) |
| 53 | \( 1 + 9.54T + 53T^{2} \) |
| 59 | \( 1 + 10.1iT - 59T^{2} \) |
| 61 | \( 1 + 12.7iT - 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 + 7.38T + 71T^{2} \) |
| 73 | \( 1 + 5.73iT - 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 - 3.12T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 - 14.6iT - 97T^{2} \) |
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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
−8.207351395827349943149808728702, −7.58401419277261703262235465076, −6.66750361640006629348069286872, −6.14010778322013760542836690655, −5.41977188924641240537574834261, −4.53882319955425640483124412398, −3.35089784297519262229814562180, −2.89627571560254719990371712938, −2.00087564329962644041241992455, −0.36840820913605070629686888001,
1.18461472018707545863954027498, 1.71272937899862618183798961694, 3.36909914296491241335454253064, 4.09311767996007059694288021151, 4.50513000073318467049198457113, 5.60971957404238509470759974754, 6.21694033598886159504813954938, 7.24521484907559934080439193053, 7.73445985203117401315272976454, 8.525532012760756171855633330567
