L(s) = 1 | − 2-s + 3-s + 4-s − 3·5-s − 6-s + 7-s − 8-s + 9-s + 3·10-s + 3·11-s + 12-s − 4·13-s − 14-s − 3·15-s + 16-s + 3·17-s − 18-s − 3·20-s + 21-s − 3·22-s − 23-s − 24-s + 4·25-s + 4·26-s + 27-s + 28-s − 6·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s + 0.904·11-s + 0.288·12-s − 1.10·13-s − 0.267·14-s − 0.774·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.670·20-s + 0.218·21-s − 0.639·22-s − 0.208·23-s − 0.204·24-s + 4/5·25-s + 0.784·26-s + 0.192·27-s + 0.188·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.53421390324032, −12.29458672718143, −11.74617004551065, −11.45223437851182, −11.11557492559205, −10.27626231713582, −10.17170280040195, −9.336936799932691, −9.254670417847986, −8.549796582186892, −8.260348375817296, −7.598851307084920, −7.504796938669423, −7.054952750254601, −6.510454816625842, −5.815329633314113, −5.211771914109769, −4.672493448432010, −4.127817538638091, −3.584742292458917, −3.292162410660212, −2.612885681212402, −1.802599715087442, −1.552696001728066, −0.6154242802410732, 0,
0.6154242802410732, 1.552696001728066, 1.802599715087442, 2.612885681212402, 3.292162410660212, 3.584742292458917, 4.127817538638091, 4.672493448432010, 5.211771914109769, 5.815329633314113, 6.510454816625842, 7.054952750254601, 7.504796938669423, 7.598851307084920, 8.260348375817296, 8.549796582186892, 9.254670417847986, 9.336936799932691, 10.17170280040195, 10.27626231713582, 11.11557492559205, 11.45223437851182, 11.74617004551065, 12.29458672718143, 12.53421390324032