| L(s) = 1 | + 0.601·2-s + 1.54·3-s − 3.63·4-s + 0.928·6-s − 4.59·8-s − 6.61·9-s − 5.61·12-s + 23.5·13-s + 11.7·16-s − 3.98·18-s − 23·23-s − 7.09·24-s + 25·25-s + 14.1·26-s − 24.1·27-s − 42.4·29-s − 27.9·31-s + 25.4·32-s + 24.0·36-s + 36.3·39-s + 74.9·41-s − 13.8·46-s − 93.8·47-s + 18.1·48-s + 49·49-s + 15.0·50-s − 85.5·52-s + ⋯ |
| L(s) = 1 | + 0.300·2-s + 0.514·3-s − 0.909·4-s + 0.154·6-s − 0.574·8-s − 0.735·9-s − 0.468·12-s + 1.80·13-s + 0.736·16-s − 0.221·18-s − 23-s − 0.295·24-s + 25-s + 0.544·26-s − 0.892·27-s − 1.46·29-s − 0.902·31-s + 0.795·32-s + 0.668·36-s + 0.930·39-s + 1.82·41-s − 0.300·46-s − 1.99·47-s + 0.379·48-s + 0.999·49-s + 0.300·50-s − 1.64·52-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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9894807023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9894807023\) |
| \(L(2)\) |
|
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 | \( 1 - 0.601T + 4T^{2} \) |
| 3 | \( 1 - 1.54T + 9T^{2} \) |
| 5 | \( 1 - 25T^{2} \) |
| 7 | \( 1 - 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 - 23.5T + 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 29 | \( 1 + 42.4T + 841T^{2} \) |
| 31 | \( 1 + 27.9T + 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 74.9T + 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3T^{2} \) |
| 47 | \( 1 + 93.8T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 26T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 140.T + 5.04e3T^{2} \) |
| 73 | \( 1 + 56.8T + 5.32e3T^{2} \) |
| 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
−17.87908189478010385509079805331, −16.38854251263791914136378027948, −14.82823558841975961978977417582, −13.87709638210915675759205960405, −12.88326648893801043752994094227, −11.12595221769137640392299636957, −9.255683549229959107838401341587, −8.249320660797426125572295542019, −5.79533420479259215020430426704, −3.69917499700457382202237665871,
3.69917499700457382202237665871, 5.79533420479259215020430426704, 8.249320660797426125572295542019, 9.255683549229959107838401341587, 11.12595221769137640392299636957, 12.88326648893801043752994094227, 13.87709638210915675759205960405, 14.82823558841975961978977417582, 16.38854251263791914136378027948, 17.87908189478010385509079805331
