L(s) = 1 | + i·2-s − i·3-s − 4-s + (−0.707 − 2.12i)5-s + 6-s − 2i·7-s − i·8-s − 9-s + (2.12 − 0.707i)10-s − 4.24·11-s + i·12-s + 0.828i·13-s + 2·14-s + (−2.12 + 0.707i)15-s + 16-s + 6.82i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (−0.316 − 0.948i)5-s + 0.408·6-s − 0.755i·7-s − 0.353i·8-s − 0.333·9-s + (0.670 − 0.223i)10-s − 1.27·11-s + 0.288i·12-s + 0.229i·13-s + 0.534·14-s + (−0.547 + 0.182i)15-s + 0.250·16-s + 1.65i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0480295 - 0.295971i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0480295 - 0.295971i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (0.707 + 2.12i)T \) |
| 23 | \( 1 - iT \) |
good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 13 | \( 1 - 0.828iT - 13T^{2} \) |
| 17 | \( 1 - 6.82iT - 17T^{2} \) |
| 19 | \( 1 + 6.24T + 19T^{2} \) |
| 29 | \( 1 + 3.65T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 0.585iT - 37T^{2} \) |
| 41 | \( 1 + 6.82T + 41T^{2} \) |
| 43 | \( 1 + 10.2iT - 43T^{2} \) |
| 47 | \( 1 + 0.828iT - 47T^{2} \) |
| 53 | \( 1 + 10.5iT - 53T^{2} \) |
| 59 | \( 1 + 8.48T + 59T^{2} \) |
| 61 | \( 1 + 0.585T + 61T^{2} \) |
| 67 | \( 1 + 3.41iT - 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 7.65iT - 73T^{2} \) |
| 79 | \( 1 - 3.65T + 79T^{2} \) |
| 83 | \( 1 + 1.41iT - 83T^{2} \) |
| 89 | \( 1 + 9.17T + 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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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
−10.13274255174500266859779060170, −8.782120607035016072054041798666, −8.242227452356724654053338419119, −7.59037229561093858775508131553, −6.58169695763093165944208686867, −5.65294831464020631034018359279, −4.65514637526271805955202461886, −3.73045055904861672129954303485, −1.82728893650453217587657874478, −0.14945565433026806110664592980,
2.51405555827995043101920717066, 2.94488202302456260376400984208, 4.33194506087654234908927267274, 5.24179012591949861526600923519, 6.29298581092539364957621248367, 7.54742181514054021058892365220, 8.390349576715508357421841945618, 9.352916556719555667384233896954, 10.19947617206147610106442181327, 10.79302885299820121484130424615