| L(s) = 1 | + 3-s + 3.46i·7-s + 9-s + 3.46i·11-s + (−1 + 3.46i)13-s − 6·17-s − 3.46i·19-s + 3.46i·21-s + 5·25-s + 27-s + 6·29-s + 3.46i·31-s + 3.46i·33-s + 6.92i·37-s + (−1 + 3.46i)39-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.30i·7-s + 0.333·9-s + 1.04i·11-s + (−0.277 + 0.960i)13-s − 1.45·17-s − 0.794i·19-s + 0.755i·21-s + 25-s + 0.192·27-s + 1.11·29-s + 0.622i·31-s + 0.603i·33-s + 1.13i·37-s + (−0.160 + 0.554i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.30885 + 0.984469i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30885 + 0.984469i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + (1 - 3.46i)T \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 - 3.46iT - 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 - 6.92iT - 37T^{2} \) |
| 41 | \( 1 + 6.92iT - 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 3.46iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 10.3iT - 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 10.3iT - 67T^{2} \) |
| 71 | \( 1 + 3.46iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 3.46iT - 83T^{2} \) |
| 89 | \( 1 + 6.92iT - 89T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
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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
−10.75689756272077707582952716091, −9.686147721688605379665328424707, −8.935791168218723743976795167051, −8.510765237236881745588180510578, −7.08042949451288875981485634498, −6.56082946455080767729923637818, −5.07844470027483760381842639919, −4.36860633865166102426951112563, −2.75771868111859403138891802347, −2.00657706072832352684294701813,
0.860671078871469360885599871875, 2.66075473752642768838636405485, 3.73598920899971406734335314969, 4.64550640467252958945267459081, 6.00493265352164549466872659514, 6.99088465860776900368203948303, 7.86718136373754710577523334846, 8.576641639165694473556274004970, 9.562311045310686475041676405335, 10.66319121561289677400757188784
