| L(s) = 1 | + 1.81·3-s + 0.710·5-s − 7-s + 0.289·9-s + 11-s + 5.10·13-s + 1.28·15-s − 4.72·17-s − 2.57·19-s − 1.81·21-s + 1.28·23-s − 4.49·25-s − 4.91·27-s + 5.62·29-s + 4.39·31-s + 1.81·33-s − 0.710·35-s + 2.71·37-s + 9.25·39-s + 11.3·41-s + 5.25·43-s + 0.205·45-s + 1.94·47-s + 49-s − 8.57·51-s + 7.62·53-s + 0.710·55-s + ⋯ |
| L(s) = 1 | + 1.04·3-s + 0.317·5-s − 0.377·7-s + 0.0963·9-s + 0.301·11-s + 1.41·13-s + 0.332·15-s − 1.14·17-s − 0.591·19-s − 0.395·21-s + 0.268·23-s − 0.898·25-s − 0.946·27-s + 1.04·29-s + 0.788·31-s + 0.315·33-s − 0.120·35-s + 0.445·37-s + 1.48·39-s + 1.76·41-s + 0.801·43-s + 0.0306·45-s + 0.283·47-s + 0.142·49-s − 1.20·51-s + 1.04·53-s + 0.0958·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.007562473\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.007562473\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 1.81T + 3T^{2} \) |
| 5 | \( 1 - 0.710T + 5T^{2} \) |
| 13 | \( 1 - 5.10T + 13T^{2} \) |
| 17 | \( 1 + 4.72T + 17T^{2} \) |
| 19 | \( 1 + 2.57T + 19T^{2} \) |
| 23 | \( 1 - 1.28T + 23T^{2} \) |
| 29 | \( 1 - 5.62T + 29T^{2} \) |
| 31 | \( 1 - 4.39T + 31T^{2} \) |
| 37 | \( 1 - 2.71T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 - 5.25T + 43T^{2} \) |
| 47 | \( 1 - 1.94T + 47T^{2} \) |
| 53 | \( 1 - 7.62T + 53T^{2} \) |
| 59 | \( 1 - 6.59T + 59T^{2} \) |
| 61 | \( 1 - 2.52T + 61T^{2} \) |
| 67 | \( 1 - 4.54T + 67T^{2} \) |
| 71 | \( 1 - 6.71T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 - 5.83T + 83T^{2} \) |
| 89 | \( 1 - 2.50T + 89T^{2} \) |
| 97 | \( 1 + 0.710T + 97T^{2} \) |
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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
−8.330873569926197389540244983744, −7.79036956513460659737079229194, −6.64903930571365948224229358191, −6.28966655949944458899053901129, −5.43836526770661077756662444711, −4.16030688020210300404618737318, −3.83296588637086814495489525105, −2.70642099406748424081350274144, −2.20275338133834264207170198226, −0.914920817811771079429097775206,
0.914920817811771079429097775206, 2.20275338133834264207170198226, 2.70642099406748424081350274144, 3.83296588637086814495489525105, 4.16030688020210300404618737318, 5.43836526770661077756662444711, 6.28966655949944458899053901129, 6.64903930571365948224229358191, 7.79036956513460659737079229194, 8.330873569926197389540244983744
