L(s) = 1 | − 3.46i·5-s − 1.73·7-s − 6i·11-s + 5.19i·13-s − 6·17-s − 5i·19-s + 3.46·23-s − 6.99·25-s + 6.92i·29-s − 3.46·31-s + 5.99i·35-s + 1.73i·37-s − 4i·43-s + 3.46·47-s − 4·49-s + ⋯ |
L(s) = 1 | − 1.54i·5-s − 0.654·7-s − 1.80i·11-s + 1.44i·13-s − 1.45·17-s − 1.14i·19-s + 0.722·23-s − 1.39·25-s + 1.28i·29-s − 0.622·31-s + 1.01i·35-s + 0.284i·37-s − 0.609i·43-s + 0.505·47-s − 0.571·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5668625877\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5668625877\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 7 | \( 1 + 1.73T + 7T^{2} \) |
| 11 | \( 1 + 6iT - 11T^{2} \) |
| 13 | \( 1 - 5.19iT - 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 5iT - 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 - 6.92iT - 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 - 1.73iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 + 6.92iT - 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 - 12.1iT - 61T^{2} \) |
| 67 | \( 1 - 5iT - 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 7T + 73T^{2} \) |
| 79 | \( 1 + 5.19T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−8.952774364612544803326093142608, −8.599206564057439879624154011392, −7.18173200956071187532436414742, −6.48740459502075039145385782136, −5.56503337399670877144535030039, −4.74490439064127255540267277504, −3.97351412925969355109850596586, −2.82605179923579814619731568052, −1.41474294288857573765650224461, −0.20885354321755968706976410599,
2.04252804209525565536207341093, 2.84561326542897479118525896008, 3.74344345731746729832954448024, 4.76150642500062643971533485193, 5.99096607040410814367549485677, 6.57263687288255213501342879076, 7.37689922160531482214435268109, 7.84416165172247766805614126758, 9.166651964715582920921978562681, 9.941949001368524104314474968925