L(s) = 1 | + 3.24·3-s − 3.61i·5-s − 8.24·7-s + 1.56·9-s + 15.8i·11-s − 15.0·13-s − 11.7i·15-s + 10.3·17-s + (18.4 − 4.43i)19-s − 26.7·21-s + 39.6·23-s + 11.9·25-s − 24.1·27-s − 3.29·29-s + 41.8i·31-s + ⋯ |
L(s) = 1 | + 1.08·3-s − 0.723i·5-s − 1.17·7-s + 0.173·9-s + 1.43i·11-s − 1.15·13-s − 0.783i·15-s + 0.609·17-s + (0.972 − 0.233i)19-s − 1.27·21-s + 1.72·23-s + 0.477·25-s − 0.895·27-s − 0.113·29-s + 1.34i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.013688698\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.013688698\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-18.4 + 4.43i)T \) |
good | 3 | \( 1 - 3.24T + 9T^{2} \) |
| 5 | \( 1 + 3.61iT - 25T^{2} \) |
| 7 | \( 1 + 8.24T + 49T^{2} \) |
| 11 | \( 1 - 15.8iT - 121T^{2} \) |
| 13 | \( 1 + 15.0T + 169T^{2} \) |
| 17 | \( 1 - 10.3T + 289T^{2} \) |
| 23 | \( 1 - 39.6T + 529T^{2} \) |
| 29 | \( 1 + 3.29T + 841T^{2} \) |
| 31 | \( 1 - 41.8iT - 961T^{2} \) |
| 37 | \( 1 - 65.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 57.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 60.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 38.6T + 2.20e3T^{2} \) |
| 53 | \( 1 + 43.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 6.27T + 3.48e3T^{2} \) |
| 61 | \( 1 - 75.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 125.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 113. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 71.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 142. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 19.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 16.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 12.5iT - 9.40e3T^{2} \) |
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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
−9.586775713827773681965646404259, −9.105511581585022802988821765925, −8.029714419708876918478334661968, −7.33713020896673699355381274923, −6.58583827323055783633183965046, −5.18298746021609310351068686158, −4.57298983498469209495505197853, −3.16209125764452921698709400782, −2.71735142095544199965500426677, −1.21571553447014465024998387857,
0.54740023365575546687848106644, 2.52415746992709528257064185608, 3.10686033889276349040626510788, 3.64865724189761462761595889762, 5.27252480963320581005770419775, 6.14122684461831772390224547359, 7.10169643167257801322877018696, 7.72991761861678057690933581530, 8.687353113491381022434161423989, 9.449079617332365658958024375408