| L(s) = 1 | − 0.725·5-s + 13.8·7-s + 9·9-s + 20.3·11-s + 18.9·17-s + 19·19-s − 30·23-s − 24.4·25-s − 10.0·35-s − 53.8·43-s − 6.52·45-s − 86.5·47-s + 142.·49-s − 14.7·55-s − 5.12·61-s + 124.·63-s − 112.·73-s + 281.·77-s + 81·81-s − 90·83-s − 13.7·85-s − 13.7·95-s + 183.·99-s + 102·101-s + 21.7·115-s + 261.·119-s + ⋯ |
| L(s) = 1 | − 0.145·5-s + 1.97·7-s + 9-s + 1.85·11-s + 1.11·17-s + 19-s − 1.30·23-s − 0.978·25-s − 0.286·35-s − 1.25·43-s − 0.145·45-s − 1.84·47-s + 2.90·49-s − 0.268·55-s − 0.0839·61-s + 1.97·63-s − 1.53·73-s + 3.65·77-s + 81-s − 1.08·83-s − 0.161·85-s − 0.145·95-s + 1.85·99-s + 1.00·101-s + 0.189·115-s + 2.19·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.111742793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.111742793\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 - 9T^{2} \) |
| 5 | \( 1 + 0.725T + 25T^{2} \) |
| 7 | \( 1 - 13.8T + 49T^{2} \) |
| 11 | \( 1 - 20.3T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 - 18.9T + 289T^{2} \) |
| 23 | \( 1 + 30T + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 + 53.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + 86.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 5.12T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 112.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 90T + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−9.655673269666963791873207697637, −8.602405603414539888927232964383, −7.85320533508161270552148029886, −7.29001599711649008202969349806, −6.19187828276540075727027518694, −5.16723219614876756657440988069, −4.32363852995342092160394397940, −3.62831969914893079497934577170, −1.71954298559270567831099226396, −1.30036670126043883232048030226,
1.30036670126043883232048030226, 1.71954298559270567831099226396, 3.62831969914893079497934577170, 4.32363852995342092160394397940, 5.16723219614876756657440988069, 6.19187828276540075727027518694, 7.29001599711649008202969349806, 7.85320533508161270552148029886, 8.602405603414539888927232964383, 9.655673269666963791873207697637
