L(s) = 1 | − 1.56·3-s − 3.56·5-s + 7-s − 0.561·9-s − 11-s − 5.12·13-s + 5.56·15-s − 2·17-s − 3.12·19-s − 1.56·21-s + 5.56·23-s + 7.68·25-s + 5.56·27-s − 2·29-s + 6.43·31-s + 1.56·33-s − 3.56·35-s + 0.438·37-s + 8·39-s − 10·41-s − 4·43-s + 2·45-s − 10.2·47-s + 49-s + 3.12·51-s + 12.2·53-s + 3.56·55-s + ⋯ |
L(s) = 1 | − 0.901·3-s − 1.59·5-s + 0.377·7-s − 0.187·9-s − 0.301·11-s − 1.42·13-s + 1.43·15-s − 0.485·17-s − 0.716·19-s − 0.340·21-s + 1.15·23-s + 1.53·25-s + 1.07·27-s − 0.371·29-s + 1.15·31-s + 0.271·33-s − 0.602·35-s + 0.0720·37-s + 1.28·39-s − 1.56·41-s − 0.609·43-s + 0.298·45-s − 1.49·47-s + 0.142·49-s + 0.437·51-s + 1.68·53-s + 0.480·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4798952350\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4798952350\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 5 | \( 1 + 3.56T + 5T^{2} \) |
| 13 | \( 1 + 5.12T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 3.12T + 19T^{2} \) |
| 23 | \( 1 - 5.56T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 6.43T + 31T^{2} \) |
| 37 | \( 1 - 0.438T + 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 9.56T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 1.56T + 67T^{2} \) |
| 71 | \( 1 - 8.68T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 - 3.12T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 + 8.43T + 89T^{2} \) |
| 97 | \( 1 - 4.43T + 97T^{2} \) |
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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
−9.912771704160120191069389134133, −8.601136053052979109463355672582, −8.151250134273511872928750589636, −7.13302618334980237138551645998, −6.61584087810848806957308593872, −5.12165277183153592454411946206, −4.84240376778594649773757067858, −3.71066124893146077095184694162, −2.51235977597565663069194621816, −0.51171148972143446251759341972,
0.51171148972143446251759341972, 2.51235977597565663069194621816, 3.71066124893146077095184694162, 4.84240376778594649773757067858, 5.12165277183153592454411946206, 6.61584087810848806957308593872, 7.13302618334980237138551645998, 8.151250134273511872928750589636, 8.601136053052979109463355672582, 9.912771704160120191069389134133