L(s) = 1 | + i·2-s + 2.44i·3-s − 4-s + (−2 + i)5-s − 2.44·6-s + 4.44i·7-s − i·8-s − 2.99·9-s + (−1 − 2i)10-s + 4.89·11-s − 2.44i·12-s − 4i·13-s − 4.44·14-s + (−2.44 − 4.89i)15-s + 16-s − 4.89i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 1.41i·3-s − 0.5·4-s + (−0.894 + 0.447i)5-s − 0.999·6-s + 1.68i·7-s − 0.353i·8-s − 0.999·9-s + (−0.316 − 0.632i)10-s + 1.47·11-s − 0.707i·12-s − 1.10i·13-s − 1.18·14-s + (−0.632 − 1.26i)15-s + 0.250·16-s − 1.18i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.233569 - 0.989414i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.233569 - 0.989414i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 + (2 - i)T \) |
| 37 | \( 1 - iT \) |
good | 3 | \( 1 - 2.44iT - 3T^{2} \) |
| 7 | \( 1 - 4.44iT - 7T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + 4.89iT - 17T^{2} \) |
| 19 | \( 1 + 3.55T + 19T^{2} \) |
| 23 | \( 1 - 8.89iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 1.55T + 31T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 4.44iT - 47T^{2} \) |
| 53 | \( 1 - 11.7iT - 53T^{2} \) |
| 59 | \( 1 - 3.55T + 59T^{2} \) |
| 61 | \( 1 - 12T + 61T^{2} \) |
| 67 | \( 1 + 5.55iT - 67T^{2} \) |
| 71 | \( 1 - 4.89T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 6.44T + 79T^{2} \) |
| 83 | \( 1 - 9.55iT - 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−11.73315927642622350769076202571, −11.07936341415248186996918047476, −9.779503443908325277954978684317, −9.158470199486185831464631784296, −8.432623073049425086652630510554, −7.23217428890444396756726139354, −5.95292952459366795557344906314, −5.12846248971825754738029212440, −4.00228210919911626960026921576, −3.04726820625125862859212933148,
0.74264998369890049481600485601, 1.78930735016282257625297775982, 3.90734779431379043345770183706, 4.30837356157822123939895533126, 6.55394008362429058527231465249, 6.94772658743868467461326405431, 8.126851331913683768832510524418, 8.780027250126575614214292812127, 10.17220101306183106204817417116, 11.18886339174866365554162552238