L(s) = 1 | + (−0.809 + 0.587i)2-s + (−0.809 − 0.587i)3-s + (0.309 − 0.951i)4-s + 5-s + 6-s + (−0.530 + 1.63i)7-s + (0.309 + 0.951i)8-s + (0.309 + 0.951i)9-s + (−0.809 + 0.587i)10-s + (0.890 − 2.74i)11-s + (−0.809 + 0.587i)12-s + (0.180 + 0.131i)13-s + (−0.530 − 1.63i)14-s + (−0.809 − 0.587i)15-s + (−0.809 − 0.587i)16-s + (1.60 + 4.93i)17-s + ⋯ |
L(s) = 1 | + (−0.572 + 0.415i)2-s + (−0.467 − 0.339i)3-s + (0.154 − 0.475i)4-s + 0.447·5-s + 0.408·6-s + (−0.200 + 0.617i)7-s + (0.109 + 0.336i)8-s + (0.103 + 0.317i)9-s + (−0.255 + 0.185i)10-s + (0.268 − 0.826i)11-s + (−0.233 + 0.169i)12-s + (0.0501 + 0.0364i)13-s + (−0.141 − 0.436i)14-s + (−0.208 − 0.151i)15-s + (−0.202 − 0.146i)16-s + (0.388 + 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.953817 + 0.417676i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.953817 + 0.417676i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 - 0.587i)T \) |
| 3 | \( 1 + (0.809 + 0.587i)T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 + (-0.731 + 5.51i)T \) |
good | 7 | \( 1 + (0.530 - 1.63i)T + (-5.66 - 4.11i)T^{2} \) |
| 11 | \( 1 + (-0.890 + 2.74i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-0.180 - 0.131i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.60 - 4.93i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.261 + 0.189i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.871 - 2.68i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (3.51 - 2.55i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 - 2.36T + 37T^{2} \) |
| 41 | \( 1 + (4.56 - 3.31i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-5.84 + 4.24i)T + (13.2 - 40.8i)T^{2} \) |
| 47 | \( 1 + (-6.83 - 4.96i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-1.25 - 3.84i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-4.48 - 3.25i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 - 12.0T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 + (-2.26 - 6.96i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (1.54 - 4.74i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (1.72 + 5.29i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (2.25 - 1.63i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (1.46 - 4.49i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (2.06 - 6.34i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.09901301965547762428124844895, −9.245698822949870746549113844539, −8.539338971028933890194361619909, −7.67924371374863493139258926118, −6.68450602182892723142451650375, −5.84588942469552409941098570460, −5.48395229058110349457903399363, −3.89474963724154432898868021104, −2.42469937613752647610905008373, −1.12791020451072619384022934070,
0.77277123078316267929624636857, 2.25204062676003263024850988946, 3.54734467847413854785211421903, 4.57816941127317523261944533188, 5.54557093307452913843564671158, 6.80344864838579405425424887009, 7.26591383142632669577297907067, 8.477693559931443049160873155187, 9.462340332244904749157219015242, 9.918067254066127251630485764732