| L(s) = 1 | + 1.81·2-s + 1.28·4-s + 1.10·5-s + 2.52·7-s − 1.28·8-s + 2·10-s − 0.813·11-s + 5.10·13-s + 4.57·14-s − 4.91·16-s − 3.39·17-s + 3.10·19-s + 1.42·20-s − 1.47·22-s + 0.897·23-s − 3.78·25-s + 9.25·26-s + 3.25·28-s − 4.44·29-s − 8.96·31-s − 6.33·32-s − 6.15·34-s + 2.78·35-s + 2.08·37-s + 5.62·38-s − 1.42·40-s − 41-s + ⋯ |
| L(s) = 1 | + 1.28·2-s + 0.644·4-s + 0.493·5-s + 0.954·7-s − 0.455·8-s + 0.632·10-s − 0.245·11-s + 1.41·13-s + 1.22·14-s − 1.22·16-s − 0.822·17-s + 0.711·19-s + 0.317·20-s − 0.314·22-s + 0.187·23-s − 0.756·25-s + 1.81·26-s + 0.615·28-s − 0.824·29-s − 1.61·31-s − 1.12·32-s − 1.05·34-s + 0.470·35-s + 0.342·37-s + 0.912·38-s − 0.224·40-s − 0.156·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.719601848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.719601848\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 1.81T + 2T^{2} \) |
| 5 | \( 1 - 1.10T + 5T^{2} \) |
| 7 | \( 1 - 2.52T + 7T^{2} \) |
| 11 | \( 1 + 0.813T + 11T^{2} \) |
| 13 | \( 1 - 5.10T + 13T^{2} \) |
| 17 | \( 1 + 3.39T + 17T^{2} \) |
| 19 | \( 1 - 3.10T + 19T^{2} \) |
| 23 | \( 1 - 0.897T + 23T^{2} \) |
| 29 | \( 1 + 4.44T + 29T^{2} \) |
| 31 | \( 1 + 8.96T + 31T^{2} \) |
| 37 | \( 1 - 2.08T + 37T^{2} \) |
| 43 | \( 1 - 9.07T + 43T^{2} \) |
| 47 | \( 1 - 0.235T + 47T^{2} \) |
| 53 | \( 1 + 13.8T + 53T^{2} \) |
| 59 | \( 1 + 1.04T + 59T^{2} \) |
| 61 | \( 1 - 1.91T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 - 4.75T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 - 7.68T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 - 2.72T + 97T^{2} \) |
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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
−11.26079070565022669820038664507, −11.09488318564911399631862582268, −9.498260073919924816635499750861, −8.659203346773457221606076024030, −7.47420489502605161507288161630, −6.15493705362680638621081274444, −5.49405308312498543391105501176, −4.47088613346130716474148893166, −3.43468880459507507479454388368, −1.90274429162386272274634479186,
1.90274429162386272274634479186, 3.43468880459507507479454388368, 4.47088613346130716474148893166, 5.49405308312498543391105501176, 6.15493705362680638621081274444, 7.47420489502605161507288161630, 8.659203346773457221606076024030, 9.498260073919924816635499750861, 11.09488318564911399631862582268, 11.26079070565022669820038664507
