| L(s) = 1 | − 1.74·2-s − 3-s + 1.03·4-s + 5-s + 1.74·6-s − 4.51·7-s + 1.67·8-s + 9-s − 1.74·10-s + 2.06·11-s − 1.03·12-s + 1.55·13-s + 7.86·14-s − 15-s − 4.99·16-s − 6.32·17-s − 1.74·18-s + 0.647·19-s + 1.03·20-s + 4.51·21-s − 3.60·22-s + 1.70·23-s − 1.67·24-s + 25-s − 2.71·26-s − 27-s − 4.68·28-s + ⋯ |
| L(s) = 1 | − 1.23·2-s − 0.577·3-s + 0.518·4-s + 0.447·5-s + 0.711·6-s − 1.70·7-s + 0.593·8-s + 0.333·9-s − 0.551·10-s + 0.622·11-s − 0.299·12-s + 0.432·13-s + 2.10·14-s − 0.258·15-s − 1.24·16-s − 1.53·17-s − 0.410·18-s + 0.148·19-s + 0.231·20-s + 0.985·21-s − 0.767·22-s + 0.355·23-s − 0.342·24-s + 0.200·25-s − 0.532·26-s − 0.192·27-s − 0.884·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4648403840\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4648403840\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 + 1.74T + 2T^{2} \) |
| 7 | \( 1 + 4.51T + 7T^{2} \) |
| 11 | \( 1 - 2.06T + 11T^{2} \) |
| 13 | \( 1 - 1.55T + 13T^{2} \) |
| 17 | \( 1 + 6.32T + 17T^{2} \) |
| 19 | \( 1 - 0.647T + 19T^{2} \) |
| 23 | \( 1 - 1.70T + 23T^{2} \) |
| 29 | \( 1 + 5.75T + 29T^{2} \) |
| 31 | \( 1 - 8.19T + 31T^{2} \) |
| 37 | \( 1 + 5.91T + 37T^{2} \) |
| 41 | \( 1 + 6.63T + 41T^{2} \) |
| 43 | \( 1 - 1.84T + 43T^{2} \) |
| 47 | \( 1 + 1.03T + 47T^{2} \) |
| 53 | \( 1 - 0.403T + 53T^{2} \) |
| 59 | \( 1 - 4.56T + 59T^{2} \) |
| 61 | \( 1 - 8.02T + 61T^{2} \) |
| 67 | \( 1 - 5.20T + 67T^{2} \) |
| 71 | \( 1 - 6.09T + 71T^{2} \) |
| 73 | \( 1 + 4.24T + 73T^{2} \) |
| 83 | \( 1 + 2.13T + 83T^{2} \) |
| 89 | \( 1 + 7.81T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−9.748887432084051949004045044079, −9.066704278632350699248157713355, −8.490024838304059236715800290204, −7.04152291338571169928071300111, −6.71346530987832929513953733859, −5.88690234148170335242495687877, −4.60453421839917394991572217764, −3.49290994150941390082422458007, −2.05775890657611219463116492001, −0.62040455439365628969312915726,
0.62040455439365628969312915726, 2.05775890657611219463116492001, 3.49290994150941390082422458007, 4.60453421839917394991572217764, 5.88690234148170335242495687877, 6.71346530987832929513953733859, 7.04152291338571169928071300111, 8.490024838304059236715800290204, 9.066704278632350699248157713355, 9.748887432084051949004045044079
