| L(s) = 1 | + 2.44i·3-s + 3.85·5-s + (−2.49 − 0.886i)7-s − 2.96·9-s + 1.81·11-s − 13-s + 9.40i·15-s + 3.46i·17-s + 4.50i·19-s + (2.16 − 6.08i)21-s + 0.813i·23-s + 9.83·25-s + 0.0798i·27-s − 2.95i·29-s − 7.74·31-s + ⋯ |
| L(s) = 1 | + 1.41i·3-s + 1.72·5-s + (−0.942 − 0.335i)7-s − 0.989·9-s + 0.546·11-s − 0.277·13-s + 2.42i·15-s + 0.839i·17-s + 1.03i·19-s + (0.472 − 1.32i)21-s + 0.169i·23-s + 1.96·25-s + 0.0153i·27-s − 0.549i·29-s − 1.39·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.687 - 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.072331885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.072331885\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.49 + 0.886i)T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2.44iT - 3T^{2} \) |
| 5 | \( 1 - 3.85T + 5T^{2} \) |
| 11 | \( 1 - 1.81T + 11T^{2} \) |
| 17 | \( 1 - 3.46iT - 17T^{2} \) |
| 19 | \( 1 - 4.50iT - 19T^{2} \) |
| 23 | \( 1 - 0.813iT - 23T^{2} \) |
| 29 | \( 1 + 2.95iT - 29T^{2} \) |
| 31 | \( 1 + 7.74T + 31T^{2} \) |
| 37 | \( 1 + 0.986iT - 37T^{2} \) |
| 41 | \( 1 - 3.45iT - 41T^{2} \) |
| 43 | \( 1 + 5.13T + 43T^{2} \) |
| 47 | \( 1 - 12.5T + 47T^{2} \) |
| 53 | \( 1 - 1.21iT - 53T^{2} \) |
| 59 | \( 1 - 12.8iT - 59T^{2} \) |
| 61 | \( 1 + 4.69T + 61T^{2} \) |
| 67 | \( 1 - 2.12T + 67T^{2} \) |
| 71 | \( 1 - 3.99iT - 71T^{2} \) |
| 73 | \( 1 - 9.25iT - 73T^{2} \) |
| 79 | \( 1 - 11.8iT - 79T^{2} \) |
| 83 | \( 1 + 17.6iT - 83T^{2} \) |
| 89 | \( 1 - 0.651iT - 89T^{2} \) |
| 97 | \( 1 - 10.3iT - 97T^{2} \) |
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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
−9.237285634259864142082844221563, −8.792517312585363118053782562146, −7.45921077347927423320289588704, −6.47123995903467204092864586962, −5.85417157649742183266084463223, −5.34051506510781979743958061609, −4.20175049634948669610207098231, −3.63300832301343378219063949310, −2.59493095629427049241255571345, −1.47436792954783208183760608498,
0.62907907517034221351934672243, 1.82163000498080711372117224316, 2.39861591436398034590259478323, 3.28324363551168003344290170533, 4.90102768024174758523174437329, 5.69872434963407708167004442438, 6.30952186064886984291454065545, 6.91411928612519299484400087605, 7.35206529368157983684113598540, 8.689575813559155849160464516403
