L(s) = 1 | − 2i·2-s + (−4.96 − 1.51i)3-s − 4·4-s − 17.2·5-s + (−3.03 + 9.93i)6-s − 11.5i·7-s + 8i·8-s + (22.3 + 15.0i)9-s + 34.5i·10-s − 2.12·11-s + (19.8 + 6.07i)12-s + 49.7·13-s − 23.1·14-s + (85.8 + 26.2i)15-s + 16·16-s − 2.70·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (−0.956 − 0.292i)3-s − 0.5·4-s − 1.54·5-s + (−0.206 + 0.676i)6-s − 0.624i·7-s + 0.353i·8-s + (0.829 + 0.559i)9-s + 1.09i·10-s − 0.0582·11-s + (0.478 + 0.146i)12-s + 1.06·13-s − 0.441·14-s + (1.47 + 0.451i)15-s + 0.250·16-s − 0.0385·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 - 0.439i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.898 - 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.482850 + 0.111899i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.482850 + 0.111899i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2iT \) |
| 3 | \( 1 + (4.96 + 1.51i)T \) |
| 23 | \( 1 + (75.3 + 80.5i)T \) |
good | 5 | \( 1 + 17.2T + 125T^{2} \) |
| 7 | \( 1 + 11.5iT - 343T^{2} \) |
| 11 | \( 1 + 2.12T + 1.33e3T^{2} \) |
| 13 | \( 1 - 49.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 2.70T + 4.91e3T^{2} \) |
| 19 | \( 1 - 134. iT - 6.85e3T^{2} \) |
| 29 | \( 1 - 125. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 92.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 118. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 272. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 148. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 268. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 490.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 159. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 584. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 719. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 268. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 628.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.33e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 1.43e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 16.8T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.67e3iT - 9.12e5T^{2} \) |
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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
−12.40192553433204285146329119737, −11.79510387428251197596928376563, −10.91222789616705009811049054598, −10.22013426011950442687480741693, −8.388601248945424095067326428002, −7.56085811871275621018209327274, −6.18637454785374385377463732383, −4.53245140557443188645535822169, −3.64676761841634607577370363649, −1.12885340670464866349299653843,
0.35235930524626964652232393238, 3.71641746529964077940071736655, 4.79183062720481905767507151452, 6.05019394193759909132861606683, 7.16663601208798866353049279594, 8.259647184295898725893954884511, 9.346475717853663163219110991350, 10.89393959994501880677149104322, 11.59549585717329732424734542933, 12.44032004414574614026683206039