| L(s) = 1 | − 2-s − 2.89·3-s + 4-s − 3.35·5-s + 2.89·6-s − 7-s − 8-s + 5.40·9-s + 3.35·10-s + 5.34·11-s − 2.89·12-s + 2.77·13-s + 14-s + 9.71·15-s + 16-s − 1.18·17-s − 5.40·18-s − 8.21·19-s − 3.35·20-s + 2.89·21-s − 5.34·22-s + 0.111·23-s + 2.89·24-s + 6.22·25-s − 2.77·26-s − 6.98·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.67·3-s + 0.5·4-s − 1.49·5-s + 1.18·6-s − 0.377·7-s − 0.353·8-s + 1.80·9-s + 1.05·10-s + 1.61·11-s − 0.837·12-s + 0.770·13-s + 0.267·14-s + 2.50·15-s + 0.250·16-s − 0.286·17-s − 1.27·18-s − 1.88·19-s − 0.749·20-s + 0.632·21-s − 1.14·22-s + 0.0231·23-s + 0.591·24-s + 1.24·25-s − 0.544·26-s − 1.34·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 2.89T + 3T^{2} \) |
| 5 | \( 1 + 3.35T + 5T^{2} \) |
| 11 | \( 1 - 5.34T + 11T^{2} \) |
| 13 | \( 1 - 2.77T + 13T^{2} \) |
| 17 | \( 1 + 1.18T + 17T^{2} \) |
| 19 | \( 1 + 8.21T + 19T^{2} \) |
| 23 | \( 1 - 0.111T + 23T^{2} \) |
| 29 | \( 1 + 7.47T + 29T^{2} \) |
| 31 | \( 1 + 3.87T + 31T^{2} \) |
| 37 | \( 1 - 1.22T + 37T^{2} \) |
| 41 | \( 1 + 4.76T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + 3.72T + 47T^{2} \) |
| 53 | \( 1 - 3.45T + 53T^{2} \) |
| 59 | \( 1 - 7.41T + 59T^{2} \) |
| 61 | \( 1 + 0.674T + 61T^{2} \) |
| 67 | \( 1 + 8.63T + 67T^{2} \) |
| 71 | \( 1 + 4.81T + 71T^{2} \) |
| 73 | \( 1 - 3.52T + 73T^{2} \) |
| 79 | \( 1 + 1.08T + 79T^{2} \) |
| 83 | \( 1 - 0.907T + 83T^{2} \) |
| 89 | \( 1 - 0.159T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−7.43886410289722890335098681476, −7.07882271337180170077788049212, −6.18877282442486042906503810404, −6.06232252738024083060233443069, −4.73976617044341742183535528919, −4.01308393959597797565908873168, −3.61840323652783675062567444410, −1.88770205462522288304191011366, −0.820984488872521232690275571059, 0,
0.820984488872521232690275571059, 1.88770205462522288304191011366, 3.61840323652783675062567444410, 4.01308393959597797565908873168, 4.73976617044341742183535528919, 6.06232252738024083060233443069, 6.18877282442486042906503810404, 7.07882271337180170077788049212, 7.43886410289722890335098681476
