| L(s) = 1 | + 5-s + 3.37·7-s + 2.37·11-s − 3.37·13-s − 6.74·17-s + 19-s + 5.37·23-s + 25-s + 2.37·29-s + 11.1·31-s + 3.37·35-s + 6·37-s + 0.255·41-s − 4.74·43-s − 9.37·47-s + 4.37·49-s + 10.1·53-s + 2.37·55-s + 5·59-s + 12.7·61-s − 3.37·65-s − 0.744·67-s − 4.37·71-s + 14.7·73-s + 8·77-s − 2.74·79-s − 10·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.27·7-s + 0.715·11-s − 0.935·13-s − 1.63·17-s + 0.229·19-s + 1.12·23-s + 0.200·25-s + 0.440·29-s + 1.99·31-s + 0.570·35-s + 0.986·37-s + 0.0398·41-s − 0.723·43-s − 1.36·47-s + 0.624·49-s + 1.38·53-s + 0.319·55-s + 0.650·59-s + 1.63·61-s − 0.418·65-s − 0.0909·67-s − 0.518·71-s + 1.72·73-s + 0.911·77-s − 0.308·79-s − 1.09·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.442532365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.442532365\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 7 | \( 1 - 3.37T + 7T^{2} \) |
| 11 | \( 1 - 2.37T + 11T^{2} \) |
| 13 | \( 1 + 3.37T + 13T^{2} \) |
| 17 | \( 1 + 6.74T + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 5.37T + 23T^{2} \) |
| 29 | \( 1 - 2.37T + 29T^{2} \) |
| 31 | \( 1 - 11.1T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 0.255T + 41T^{2} \) |
| 43 | \( 1 + 4.74T + 43T^{2} \) |
| 47 | \( 1 + 9.37T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 - 5T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 + 0.744T + 67T^{2} \) |
| 71 | \( 1 + 4.37T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 + 2.74T + 79T^{2} \) |
| 83 | \( 1 + 10T + 83T^{2} \) |
| 89 | \( 1 + 4.37T + 89T^{2} \) |
| 97 | \( 1 + 4.74T + 97T^{2} \) |
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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
−8.519795717253944912663599767676, −8.127776229435482206257163639049, −6.93233445152459762775167582449, −6.65707135671478672315207342489, −5.46860093726808454451869375991, −4.73846595381922484920980904226, −4.27180001033612376583202349250, −2.82634096317416370562432484461, −2.05625676713037649228336489776, −0.987532938810142313325817271311,
0.987532938810142313325817271311, 2.05625676713037649228336489776, 2.82634096317416370562432484461, 4.27180001033612376583202349250, 4.73846595381922484920980904226, 5.46860093726808454451869375991, 6.65707135671478672315207342489, 6.93233445152459762775167582449, 8.127776229435482206257163639049, 8.519795717253944912663599767676
