L(s) = 1 | − 3-s + 5-s − 1.56·7-s + 9-s − 3.12·11-s + 2·13-s − 15-s + 3.56·17-s − 2·19-s + 1.56·21-s + 23-s + 25-s − 27-s + 6.68·29-s + 4.68·31-s + 3.12·33-s − 1.56·35-s + 2.43·37-s − 2·39-s − 2.68·41-s + 45-s + 4·47-s − 4.56·49-s − 3.56·51-s + 7.56·53-s − 3.12·55-s + 2·57-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.590·7-s + 0.333·9-s − 0.941·11-s + 0.554·13-s − 0.258·15-s + 0.863·17-s − 0.458·19-s + 0.340·21-s + 0.208·23-s + 0.200·25-s − 0.192·27-s + 1.24·29-s + 0.841·31-s + 0.543·33-s − 0.263·35-s + 0.400·37-s − 0.320·39-s − 0.419·41-s + 0.149·45-s + 0.583·47-s − 0.651·49-s − 0.498·51-s + 1.03·53-s − 0.421·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.333433424\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.333433424\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 + 1.56T + 7T^{2} \) |
| 11 | \( 1 + 3.12T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 3.56T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 29 | \( 1 - 6.68T + 29T^{2} \) |
| 31 | \( 1 - 4.68T + 31T^{2} \) |
| 37 | \( 1 - 2.43T + 37T^{2} \) |
| 41 | \( 1 + 2.68T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 7.56T + 53T^{2} \) |
| 59 | \( 1 - 3.56T + 59T^{2} \) |
| 61 | \( 1 - 9.12T + 61T^{2} \) |
| 67 | \( 1 - 8.68T + 67T^{2} \) |
| 71 | \( 1 - 0.438T + 71T^{2} \) |
| 73 | \( 1 + 4.24T + 73T^{2} \) |
| 79 | \( 1 + 2.87T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 + 5.12T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−9.909371814498000134680476353932, −8.749046960048740532983138058577, −8.005839138507561840374239016786, −6.96442997931568443868765529693, −6.22903434804605607914680826998, −5.51054429369788833015189081949, −4.65340904714962494252526533382, −3.44412853848907947352865275129, −2.40372412415604193649867356248, −0.868346963445378132309991447424,
0.868346963445378132309991447424, 2.40372412415604193649867356248, 3.44412853848907947352865275129, 4.65340904714962494252526533382, 5.51054429369788833015189081949, 6.22903434804605607914680826998, 6.96442997931568443868765529693, 8.005839138507561840374239016786, 8.749046960048740532983138058577, 9.909371814498000134680476353932