| L(s) = 1 | + (0.975 − 2.13i)2-s + (−2.16 − 0.635i)3-s + (−2.30 − 2.65i)4-s + (0.841 − 0.540i)5-s + (−3.46 + 4.00i)6-s + (−0.0703 + 0.489i)7-s + (−3.42 + 1.00i)8-s + (1.75 + 1.12i)9-s + (−0.334 − 2.32i)10-s + (0.927 + 2.03i)11-s + (3.29 + 7.21i)12-s + (−0.649 − 4.51i)13-s + (0.977 + 0.628i)14-s + (−2.16 + 0.635i)15-s + (−0.190 + 1.32i)16-s + (4.51 − 5.20i)17-s + ⋯ |
| L(s) = 1 | + (0.690 − 1.51i)2-s + (−1.24 − 0.366i)3-s + (−1.15 − 1.32i)4-s + (0.376 − 0.241i)5-s + (−1.41 + 1.63i)6-s + (−0.0266 + 0.185i)7-s + (−1.20 + 0.355i)8-s + (0.584 + 0.375i)9-s + (−0.105 − 0.735i)10-s + (0.279 + 0.612i)11-s + (0.951 + 2.08i)12-s + (−0.180 − 1.25i)13-s + (0.261 + 0.167i)14-s + (−0.558 + 0.164i)15-s + (−0.0476 + 0.331i)16-s + (1.09 − 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 + 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.203641 - 0.998007i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.203641 - 0.998007i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.841 + 0.540i)T \) |
| 23 | \( 1 + (0.243 - 4.78i)T \) |
| good | 2 | \( 1 + (-0.975 + 2.13i)T + (-1.30 - 1.51i)T^{2} \) |
| 3 | \( 1 + (2.16 + 0.635i)T + (2.52 + 1.62i)T^{2} \) |
| 7 | \( 1 + (0.0703 - 0.489i)T + (-6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (-0.927 - 2.03i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (0.649 + 4.51i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-4.51 + 5.20i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-3.70 - 4.27i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (6.37 - 7.35i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-4.07 + 1.19i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (-1.68 - 1.08i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (1.84 - 1.18i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (5.71 + 1.67i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + 3.87T + 47T^{2} \) |
| 53 | \( 1 + (1.21 - 8.44i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (0.417 + 2.90i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-6.64 + 1.95i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (1.58 - 3.47i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-1.14 + 2.51i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-7.21 - 8.33i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (2.12 + 14.7i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (7.90 + 5.07i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-4.45 - 1.30i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (6.88 - 4.42i)T + (40.2 - 88.2i)T^{2} \) |
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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
−12.65750395214878863511471378628, −12.09626612396659247279461909632, −11.41111830455740686951492276331, −10.26073946216490009247490030645, −9.557666840176119151402229449286, −7.43246088522465277259802477511, −5.62412915198881838474775638050, −5.12076488149929217855842816388, −3.23521751718445555285196398769, −1.25460227241588193793356784935,
4.04933865508154266174529731721, 5.23926116386494329757070911497, 6.13097650091592773376196465497, 6.87041420211337842555458285959, 8.302127323268064904678729950544, 9.786866105377199997241415865329, 11.06736582427863975113224011482, 12.04466666077026200311471557691, 13.33651264040116810489423526619, 14.17737703090950475785949135033
