| L(s) = 1 | + 5-s + 7-s − 4.82·11-s + 0.828·13-s + 7.65·17-s + 2.82·19-s + 3.65·23-s + 25-s + 8·29-s − 8.48·31-s + 35-s − 6·37-s − 7.65·41-s − 1.65·43-s + 4·47-s + 49-s + 5.17·53-s − 4.82·55-s + 4·59-s + 6·61-s + 0.828·65-s + 15.3·67-s − 10.4·71-s − 12.1·73-s − 4.82·77-s − 5.65·79-s + 8·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 1.45·11-s + 0.229·13-s + 1.85·17-s + 0.648·19-s + 0.762·23-s + 0.200·25-s + 1.48·29-s − 1.52·31-s + 0.169·35-s − 0.986·37-s − 1.19·41-s − 0.252·43-s + 0.583·47-s + 0.142·49-s + 0.710·53-s − 0.651·55-s + 0.520·59-s + 0.768·61-s + 0.102·65-s + 1.87·67-s − 1.24·71-s − 1.42·73-s − 0.550·77-s − 0.636·79-s + 0.878·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.247120215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.247120215\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 - 0.828T + 13T^{2} \) |
| 17 | \( 1 - 7.65T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 3.65T + 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 + 8.48T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 7.65T + 41T^{2} \) |
| 43 | \( 1 + 1.65T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 5.17T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 15.3T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 + 5.65T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 + 5.31T + 89T^{2} \) |
| 97 | \( 1 + 3.17T + 97T^{2} \) |
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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
−8.293642887856621954433026166280, −7.45278080580551759460072656153, −7.01316610418863208962951195280, −5.78686967982184528124523203960, −5.37546549144402209424307199718, −4.81546167424640152249029427869, −3.51472111741619321192612622738, −2.93276979399372024108144026770, −1.88355862861168605886959550187, −0.839728657313364732239985179111,
0.839728657313364732239985179111, 1.88355862861168605886959550187, 2.93276979399372024108144026770, 3.51472111741619321192612622738, 4.81546167424640152249029427869, 5.37546549144402209424307199718, 5.78686967982184528124523203960, 7.01316610418863208962951195280, 7.45278080580551759460072656153, 8.293642887856621954433026166280
