| L(s) = 1 | − i·2-s + i·3-s − 4-s + (0.707 + 2.12i)5-s + 6-s − 2.41i·7-s + i·8-s − 9-s + (2.12 − 0.707i)10-s − 6.41·11-s − i·12-s − 5.24i·13-s − 2.41·14-s + (−2.12 + 0.707i)15-s + 16-s + 3.58i·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.316 + 0.948i)5-s + 0.408·6-s − 0.912i·7-s + 0.353i·8-s − 0.333·9-s + (0.670 − 0.223i)10-s − 1.93·11-s − 0.288i·12-s − 1.45i·13-s − 0.645·14-s + (−0.547 + 0.182i)15-s + 0.250·16-s + 0.869i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4198620739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4198620739\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-0.707 - 2.12i)T \) |
| 37 | \( 1 + iT \) |
| good | 7 | \( 1 + 2.41iT - 7T^{2} \) |
| 11 | \( 1 + 6.41T + 11T^{2} \) |
| 13 | \( 1 + 5.24iT - 13T^{2} \) |
| 17 | \( 1 - 3.58iT - 17T^{2} \) |
| 19 | \( 1 - 2.17T + 19T^{2} \) |
| 23 | \( 1 + 3iT - 23T^{2} \) |
| 29 | \( 1 + 7.07T + 29T^{2} \) |
| 31 | \( 1 + 3.65T + 31T^{2} \) |
| 41 | \( 1 + 6.58T + 41T^{2} \) |
| 43 | \( 1 + 10.4iT - 43T^{2} \) |
| 47 | \( 1 + 11.8iT - 47T^{2} \) |
| 53 | \( 1 - 3.82iT - 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 + 6.48T + 61T^{2} \) |
| 67 | \( 1 + 0.828iT - 67T^{2} \) |
| 71 | \( 1 + 9.41T + 71T^{2} \) |
| 73 | \( 1 + 2.17iT - 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 + 6.89iT - 83T^{2} \) |
| 89 | \( 1 + 2.89T + 89T^{2} \) |
| 97 | \( 1 + 6.72iT - 97T^{2} \) |
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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
−10.07578728508869633560703413116, −8.739993089050823972134375580122, −7.81842300123690586617645336177, −7.20304895068573087182937466285, −5.70072068444289118259050456784, −5.23976147055750973998115521099, −3.84722069413717081328236694406, −3.17461469679749977336425181653, −2.19457354981061130362456614644, −0.17281953492235326151670769701,
1.70346025938810581967365430448, 2.82872287498152258370242058337, 4.49686991598703664509884565010, 5.38773032673114205143703896661, 5.75028243834403966390769203499, 6.99255461976536827688969365086, 7.74203092218434566120257952732, 8.466558543613646462699700530370, 9.287277963639603930108243252247, 9.754114774114758465716544464276
