| L(s) = 1 | − 3.72·2-s + 4.24·3-s + 9.87·4-s − 15.8·6-s − 21.9·8-s + 9.05·9-s + 41.9·12-s − 21.3·13-s + 42.0·16-s − 33.7·18-s − 23·23-s − 93.0·24-s + 25·25-s + 79.5·26-s + 0.244·27-s + 55.4·29-s − 33.9·31-s − 69.1·32-s + 89.4·36-s − 90.7·39-s − 8.78·41-s + 85.6·46-s + 42.8·47-s + 178.·48-s + 49·49-s − 93.1·50-s − 211.·52-s + ⋯ |
| L(s) = 1 | − 1.86·2-s + 1.41·3-s + 2.46·4-s − 2.63·6-s − 2.73·8-s + 1.00·9-s + 3.49·12-s − 1.64·13-s + 2.62·16-s − 1.87·18-s − 23-s − 3.87·24-s + 25-s + 3.06·26-s + 0.00906·27-s + 1.91·29-s − 1.09·31-s − 2.16·32-s + 2.48·36-s − 2.32·39-s − 0.214·41-s + 1.86·46-s + 0.912·47-s + 3.72·48-s + 0.999·49-s − 1.86·50-s − 4.05·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6134696799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6134696799\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + 23T \) |
| good | 2 | \( 1 + 3.72T + 4T^{2} \) |
| 3 | \( 1 - 4.24T + 9T^{2} \) |
| 5 | \( 1 - 25T^{2} \) |
| 7 | \( 1 - 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 + 21.3T + 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 29 | \( 1 - 55.4T + 841T^{2} \) |
| 31 | \( 1 + 33.9T + 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 + 8.78T + 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3T^{2} \) |
| 47 | \( 1 - 42.8T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 26T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 + 85.6T + 5.04e3T^{2} \) |
| 73 | \( 1 - 144.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−17.86858580166154762691870631547, −16.66474855141397445291880866160, −15.38379949405769162888300515781, −14.30448934002337246511345373331, −12.18652506141296656813498136593, −10.33664492004809423462870333956, −9.319600292021350002126565316276, −8.254627656874356514129115296000, −7.15878206375237697440477144189, −2.50316094689835549482408964468,
2.50316094689835549482408964468, 7.15878206375237697440477144189, 8.254627656874356514129115296000, 9.319600292021350002126565316276, 10.33664492004809423462870333956, 12.18652506141296656813498136593, 14.30448934002337246511345373331, 15.38379949405769162888300515781, 16.66474855141397445291880866160, 17.86858580166154762691870631547
