L(s) = 1 | + 1.11i·5-s − 1.19i·7-s − 3.72·11-s + 0.113·13-s − 2.89i·17-s + i·19-s − 1.35·23-s + 3.76·25-s − 0.860i·29-s + 9.01i·31-s + 1.32·35-s + 5.64·37-s + 10.5i·41-s + 2.47i·43-s + 5.05·47-s + ⋯ |
L(s) = 1 | + 0.496i·5-s − 0.451i·7-s − 1.12·11-s + 0.0314·13-s − 0.701i·17-s + 0.229i·19-s − 0.282·23-s + 0.753·25-s − 0.159i·29-s + 1.61i·31-s + 0.224·35-s + 0.927·37-s + 1.64i·41-s + 0.376i·43-s + 0.737·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0917 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0917 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.243662266\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.243662266\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - iT \) |
good | 5 | \( 1 - 1.11iT - 5T^{2} \) |
| 7 | \( 1 + 1.19iT - 7T^{2} \) |
| 11 | \( 1 + 3.72T + 11T^{2} \) |
| 13 | \( 1 - 0.113T + 13T^{2} \) |
| 17 | \( 1 + 2.89iT - 17T^{2} \) |
| 23 | \( 1 + 1.35T + 23T^{2} \) |
| 29 | \( 1 + 0.860iT - 29T^{2} \) |
| 31 | \( 1 - 9.01iT - 31T^{2} \) |
| 37 | \( 1 - 5.64T + 37T^{2} \) |
| 41 | \( 1 - 10.5iT - 41T^{2} \) |
| 43 | \( 1 - 2.47iT - 43T^{2} \) |
| 47 | \( 1 - 5.05T + 47T^{2} \) |
| 53 | \( 1 - 9.51iT - 53T^{2} \) |
| 59 | \( 1 + 8.73T + 59T^{2} \) |
| 61 | \( 1 + 8.52T + 61T^{2} \) |
| 67 | \( 1 - 0.857iT - 67T^{2} \) |
| 71 | \( 1 - 1.91T + 71T^{2} \) |
| 73 | \( 1 - 3.06T + 73T^{2} \) |
| 79 | \( 1 - 7.62iT - 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 - 13.1iT - 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−9.032371729200035984806820709917, −8.082111475263518360793136161061, −7.52446625947051597318464207526, −6.79837043120967775006374194260, −5.98160273039316019918939404279, −5.05681207748264119600299151206, −4.35014787402722000493919275386, −3.14992680362851204589400083783, −2.58949098496920594021628931068, −1.12921814110593874558285248638,
0.43827099016146855280787380230, 1.93870892432343067798098003709, 2.78044211009043130047685665023, 3.92043982149117608625070152447, 4.78487867124417079337192974376, 5.57736476527135846541524050570, 6.16117133181170938509649946619, 7.29826048557592997413361096712, 7.920278741954730877942274456192, 8.685494347319332798080619166878