| L(s) = 1 | − 1.70·2-s + 0.910·4-s + 0.663·7-s + 1.85·8-s + 4.62·11-s − 0.00742·13-s − 1.13·14-s − 4.99·16-s + 7.19·17-s + 2.48·19-s − 7.88·22-s + 1.81·23-s + 0.0126·26-s + 0.604·28-s − 2.75·29-s + 8.74·31-s + 4.79·32-s − 12.2·34-s + 37-s − 4.23·38-s + 4.82·41-s + 1.67·43-s + 4.20·44-s − 3.08·46-s + 12.6·47-s − 6.55·49-s − 0.00675·52-s + ⋯ |
| L(s) = 1 | − 1.20·2-s + 0.455·4-s + 0.250·7-s + 0.657·8-s + 1.39·11-s − 0.00205·13-s − 0.302·14-s − 1.24·16-s + 1.74·17-s + 0.569·19-s − 1.68·22-s + 0.377·23-s + 0.00248·26-s + 0.114·28-s − 0.511·29-s + 1.56·31-s + 0.848·32-s − 2.10·34-s + 0.164·37-s − 0.686·38-s + 0.753·41-s + 0.254·43-s + 0.634·44-s − 0.455·46-s + 1.84·47-s − 0.937·49-s − 0.000937·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.483657579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.483657579\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 2 | \( 1 + 1.70T + 2T^{2} \) |
| 7 | \( 1 - 0.663T + 7T^{2} \) |
| 11 | \( 1 - 4.62T + 11T^{2} \) |
| 13 | \( 1 + 0.00742T + 13T^{2} \) |
| 17 | \( 1 - 7.19T + 17T^{2} \) |
| 19 | \( 1 - 2.48T + 19T^{2} \) |
| 23 | \( 1 - 1.81T + 23T^{2} \) |
| 29 | \( 1 + 2.75T + 29T^{2} \) |
| 31 | \( 1 - 8.74T + 31T^{2} \) |
| 41 | \( 1 - 4.82T + 41T^{2} \) |
| 43 | \( 1 - 1.67T + 43T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 - 8.06T + 53T^{2} \) |
| 59 | \( 1 + 2.99T + 59T^{2} \) |
| 61 | \( 1 - 4.10T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 - 4.51T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 3.61T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 - 15.9T + 97T^{2} \) |
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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
−7.79527913713107333935450530465, −7.42974330195337793343109679019, −6.62869365149884668534210920001, −5.86572401518160359595563354247, −5.03741839091751550445875763486, −4.20169581626729240114209884327, −3.49169711971668579052444022874, −2.41255066414760367903019367966, −1.25410797349971780675882248607, −0.896576100759044872422466053791,
0.896576100759044872422466053791, 1.25410797349971780675882248607, 2.41255066414760367903019367966, 3.49169711971668579052444022874, 4.20169581626729240114209884327, 5.03741839091751550445875763486, 5.86572401518160359595563354247, 6.62869365149884668534210920001, 7.42974330195337793343109679019, 7.79527913713107333935450530465
