L(s) = 1 | + (−0.456 + 1.33i)2-s + (−0.784 − 1.54i)3-s + (−1.58 − 1.22i)4-s + 1.42i·5-s + (2.42 − 0.344i)6-s + 1.12i·7-s + (2.35 − 1.56i)8-s + (−1.77 + 2.42i)9-s + (−1.90 − 0.649i)10-s + 1.68·11-s + (−0.644 + 3.40i)12-s + 5.14·13-s + (−1.50 − 0.513i)14-s + (2.19 − 1.11i)15-s + (1.01 + 3.86i)16-s + 4.65i·17-s + ⋯ |
L(s) = 1 | + (−0.322 + 0.946i)2-s + (−0.452 − 0.891i)3-s + (−0.791 − 0.610i)4-s + 0.636i·5-s + (0.990 − 0.140i)6-s + 0.425i·7-s + (0.833 − 0.552i)8-s + (−0.590 + 0.807i)9-s + (−0.602 − 0.205i)10-s + 0.507·11-s + (−0.186 + 0.982i)12-s + 1.42·13-s + (−0.402 − 0.137i)14-s + (0.567 − 0.288i)15-s + (0.253 + 0.967i)16-s + 1.12i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.186 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.186 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.670730 + 0.555573i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.670730 + 0.555573i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.456 - 1.33i)T \) |
| 3 | \( 1 + (0.784 + 1.54i)T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 1.42iT - 5T^{2} \) |
| 7 | \( 1 - 1.12iT - 7T^{2} \) |
| 11 | \( 1 - 1.68T + 11T^{2} \) |
| 13 | \( 1 - 5.14T + 13T^{2} \) |
| 17 | \( 1 - 4.65iT - 17T^{2} \) |
| 19 | \( 1 - 5.86iT - 19T^{2} \) |
| 29 | \( 1 - 3.67iT - 29T^{2} \) |
| 31 | \( 1 + 3.79iT - 31T^{2} \) |
| 37 | \( 1 + 2.52T + 37T^{2} \) |
| 41 | \( 1 + 8.23iT - 41T^{2} \) |
| 43 | \( 1 + 3.72iT - 43T^{2} \) |
| 47 | \( 1 + 7.62T + 47T^{2} \) |
| 53 | \( 1 - 2.23iT - 53T^{2} \) |
| 59 | \( 1 - 3.98T + 59T^{2} \) |
| 61 | \( 1 + 7.14T + 61T^{2} \) |
| 67 | \( 1 - 12.2iT - 67T^{2} \) |
| 71 | \( 1 - 3.55T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 + 5.27iT - 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 + 15.9iT - 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−12.21335900920693945157352271741, −11.00303118761748762216432691673, −10.30883211029700891249182057020, −8.801848594493686468854084718022, −8.171350959771423385621695986078, −7.04022628917522463414554127585, −6.20809269337055899067651647298, −5.60978640272925806483330102904, −3.80100728948319075312021287785, −1.56861565431745221342654222744,
0.910407994674419831763161235914, 3.16564770983114775380089081449, 4.30142340353530956935494967346, 5.11388310240615633486660335207, 6.66429475840716951450134743389, 8.308593055904028351485106790751, 9.113004763181452587531870810229, 9.766915677963044096224951281253, 10.96333789233621828474536473299, 11.34350445206089183174877826092