L(s) = 1 | + (0.912 − 1.08i)2-s + (−1.34 + 1.34i)3-s + (−0.335 − 1.97i)4-s + (−2.21 + 0.272i)5-s + (0.225 + 2.67i)6-s + (0.697 + 0.697i)7-s + (−2.43 − 1.43i)8-s − 0.592i·9-s + (−1.73 + 2.64i)10-s + 4.74i·11-s + (3.09 + 2.19i)12-s + (3.30 + 3.30i)13-s + (1.39 − 0.117i)14-s + (2.60 − 3.33i)15-s + (−3.77 + 1.32i)16-s + (−1.03 + 1.03i)17-s + ⋯ |
L(s) = 1 | + (0.645 − 0.764i)2-s + (−0.773 + 0.773i)3-s + (−0.167 − 0.985i)4-s + (−0.992 + 0.121i)5-s + (0.0920 + 1.09i)6-s + (0.263 + 0.263i)7-s + (−0.861 − 0.507i)8-s − 0.197i·9-s + (−0.547 + 0.836i)10-s + 1.43i·11-s + (0.892 + 0.633i)12-s + (0.916 + 0.916i)13-s + (0.371 − 0.0313i)14-s + (0.673 − 0.862i)15-s + (−0.943 + 0.330i)16-s + (−0.249 + 0.249i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.259 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.259 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.708740 + 0.543185i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.708740 + 0.543185i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.912 + 1.08i)T \) |
| 5 | \( 1 + (2.21 - 0.272i)T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + (1.34 - 1.34i)T - 3iT^{2} \) |
| 7 | \( 1 + (-0.697 - 0.697i)T + 7iT^{2} \) |
| 11 | \( 1 - 4.74iT - 11T^{2} \) |
| 13 | \( 1 + (-3.30 - 3.30i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.03 - 1.03i)T - 17iT^{2} \) |
| 23 | \( 1 + (6.40 - 6.40i)T - 23iT^{2} \) |
| 29 | \( 1 - 1.98iT - 29T^{2} \) |
| 31 | \( 1 + 9.29iT - 31T^{2} \) |
| 37 | \( 1 + (5.70 - 5.70i)T - 37iT^{2} \) |
| 41 | \( 1 - 9.41T + 41T^{2} \) |
| 43 | \( 1 + (-1.89 + 1.89i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.10 - 1.10i)T + 47iT^{2} \) |
| 53 | \( 1 + (0.00450 + 0.00450i)T + 53iT^{2} \) |
| 59 | \( 1 + 3.02T + 59T^{2} \) |
| 61 | \( 1 + 0.724T + 61T^{2} \) |
| 67 | \( 1 + (-10.1 - 10.1i)T + 67iT^{2} \) |
| 71 | \( 1 + 9.29iT - 71T^{2} \) |
| 73 | \( 1 + (4.54 + 4.54i)T + 73iT^{2} \) |
| 79 | \( 1 - 0.319T + 79T^{2} \) |
| 83 | \( 1 + (2.15 - 2.15i)T - 83iT^{2} \) |
| 89 | \( 1 - 8.13iT - 89T^{2} \) |
| 97 | \( 1 + (8.29 - 8.29i)T - 97iT^{2} \) |
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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
−11.61501520398900764524227711009, −10.88493661696901919861423149380, −10.03869539280773313122130803177, −9.165563081658358370455155752834, −7.79127129701910468142373971155, −6.49108742731726077294018320627, −5.39556628015679222193285184737, −4.33043980624931562322606461982, −3.91464338506096298883520301129, −1.98965668816097269078463139411,
0.53701488288438477331618364115, 3.23343037540657083998074101688, 4.26333474853128881415001582604, 5.61291719010583784797945174130, 6.27987760388826868613279707996, 7.24973150649372532647446231801, 8.190983136698962038864036760352, 8.719273274782989136718020472675, 10.82082671329194091808481763934, 11.29025583162901493185914660390