| L(s) = 1 | + 0.765·5-s + 1.10·7-s + 11-s + 13-s + 3.49·17-s − 1.57·19-s − 2.72·23-s − 4.41·25-s + 4.34·29-s + 8.22·31-s + 0.848·35-s + 3.83·37-s − 0.891·41-s + 5.49·43-s − 3.62·47-s − 5.77·49-s + 2.68·53-s + 0.765·55-s + 14.1·59-s − 7.62·61-s + 0.765·65-s − 11.3·67-s + 6.93·71-s + 4.04·73-s + 1.10·77-s + 0.0371·79-s + 11.4·83-s + ⋯ |
| L(s) = 1 | + 0.342·5-s + 0.418·7-s + 0.301·11-s + 0.277·13-s + 0.847·17-s − 0.361·19-s − 0.568·23-s − 0.882·25-s + 0.806·29-s + 1.47·31-s + 0.143·35-s + 0.630·37-s − 0.139·41-s + 0.837·43-s − 0.528·47-s − 0.824·49-s + 0.368·53-s + 0.103·55-s + 1.84·59-s − 0.975·61-s + 0.0949·65-s − 1.38·67-s + 0.823·71-s + 0.473·73-s + 0.126·77-s + 0.00417·79-s + 1.25·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.381238039\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.381238039\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 0.765T + 5T^{2} \) |
| 7 | \( 1 - 1.10T + 7T^{2} \) |
| 17 | \( 1 - 3.49T + 17T^{2} \) |
| 19 | \( 1 + 1.57T + 19T^{2} \) |
| 23 | \( 1 + 2.72T + 23T^{2} \) |
| 29 | \( 1 - 4.34T + 29T^{2} \) |
| 31 | \( 1 - 8.22T + 31T^{2} \) |
| 37 | \( 1 - 3.83T + 37T^{2} \) |
| 41 | \( 1 + 0.891T + 41T^{2} \) |
| 43 | \( 1 - 5.49T + 43T^{2} \) |
| 47 | \( 1 + 3.62T + 47T^{2} \) |
| 53 | \( 1 - 2.68T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 + 7.62T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 6.93T + 71T^{2} \) |
| 73 | \( 1 - 4.04T + 73T^{2} \) |
| 79 | \( 1 - 0.0371T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 - 3.83T + 97T^{2} \) |
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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
−8.081374583920331195139542647085, −7.73251627886547797949996936645, −6.59829398024939964612981989483, −6.14451533116522925877026957976, −5.33440483738254885236735725150, −4.52259376987182307477255664800, −3.78070368605548835535541598426, −2.79143687153776965670039703580, −1.86669165335278159881904122099, −0.871644117081632509638603868817,
0.871644117081632509638603868817, 1.86669165335278159881904122099, 2.79143687153776965670039703580, 3.78070368605548835535541598426, 4.52259376987182307477255664800, 5.33440483738254885236735725150, 6.14451533116522925877026957976, 6.59829398024939964612981989483, 7.73251627886547797949996936645, 8.081374583920331195139542647085
