L(s) = 1 | − 2.90i·3-s + (−0.311 + 2.21i)5-s − 3.52i·7-s − 5.42·9-s − 3.80·11-s + 2.62i·13-s + (6.42 + 0.903i)15-s − 5.80i·17-s − 5.05·19-s − 10.2·21-s + 0.474i·23-s + (−4.80 − 1.37i)25-s + 7.05i·27-s + 2·29-s − 2.75·31-s + ⋯ |
L(s) = 1 | − 1.67i·3-s + (−0.139 + 0.990i)5-s − 1.33i·7-s − 1.80·9-s − 1.14·11-s + 0.727i·13-s + (1.65 + 0.233i)15-s − 1.40i·17-s − 1.15·19-s − 2.23·21-s + 0.0989i·23-s + (−0.961 − 0.275i)25-s + 1.35i·27-s + 0.371·29-s − 0.494·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0511521 + 0.731729i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0511521 + 0.731729i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.311 - 2.21i)T \) |
good | 3 | \( 1 + 2.90iT - 3T^{2} \) |
| 7 | \( 1 + 3.52iT - 7T^{2} \) |
| 11 | \( 1 + 3.80T + 11T^{2} \) |
| 13 | \( 1 - 2.62iT - 13T^{2} \) |
| 17 | \( 1 + 5.80iT - 17T^{2} \) |
| 19 | \( 1 + 5.05T + 19T^{2} \) |
| 23 | \( 1 - 0.474iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 2.75T + 31T^{2} \) |
| 37 | \( 1 + 7.18iT - 37T^{2} \) |
| 41 | \( 1 - 5.18T + 41T^{2} \) |
| 43 | \( 1 - 1.95iT - 43T^{2} \) |
| 47 | \( 1 - 5.33iT - 47T^{2} \) |
| 53 | \( 1 - 5.37iT - 53T^{2} \) |
| 59 | \( 1 - 5.05T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 7.76iT - 67T^{2} \) |
| 71 | \( 1 + 4.85T + 71T^{2} \) |
| 73 | \( 1 + 6.66iT - 73T^{2} \) |
| 79 | \( 1 - 5.24T + 79T^{2} \) |
| 83 | \( 1 + 12.1iT - 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 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
−10.45762675366956237927936841336, −9.154118766440421438331511800932, −7.77186176213257080314854035716, −7.48601860974560626250952058821, −6.80244844248813939282411480483, −5.99275093925646205345690649538, −4.46370220917528219894505116518, −3.01728566509559004234675024774, −2.03547597235539946391631485185, −0.37157634925857915488233245041,
2.38264388151561916408534210290, 3.64050511677064423155726932241, 4.67942895187311250777996678728, 5.37676412760819235946041196478, 6.02592051987574299632562701669, 8.132817119692115648641897581154, 8.515423330184630593255189700099, 9.259919952175720713834693714574, 10.22901116427391362902764350684, 10.70662828619456734042277109513