L(s) = 1 | + 2-s − 3-s + 4-s + 0.260·5-s − 6-s + 7-s + 8-s + 9-s + 0.260·10-s − 0.244·11-s − 12-s + 13-s + 14-s − 0.260·15-s + 16-s + 17-s + 18-s + 2.37·19-s + 0.260·20-s − 21-s − 0.244·22-s − 4.97·23-s − 24-s − 4.93·25-s + 26-s − 27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.116·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.0822·10-s − 0.0737·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.0671·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s + 0.544·19-s + 0.0581·20-s − 0.218·21-s − 0.0521·22-s − 1.03·23-s − 0.204·24-s − 0.986·25-s + 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9282 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9282 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 - 0.260T + 5T^{2} \) |
| 11 | \( 1 + 0.244T + 11T^{2} \) |
| 19 | \( 1 - 2.37T + 19T^{2} \) |
| 23 | \( 1 + 4.97T + 23T^{2} \) |
| 29 | \( 1 + 2.80T + 29T^{2} \) |
| 31 | \( 1 + 8.78T + 31T^{2} \) |
| 37 | \( 1 + 5.58T + 37T^{2} \) |
| 41 | \( 1 + 7.51T + 41T^{2} \) |
| 43 | \( 1 - 9.15T + 43T^{2} \) |
| 47 | \( 1 + 5.23T + 47T^{2} \) |
| 53 | \( 1 - 3.82T + 53T^{2} \) |
| 59 | \( 1 - 2.72T + 59T^{2} \) |
| 61 | \( 1 + 12.8T + 61T^{2} \) |
| 67 | \( 1 + 3.08T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 - 7.03T + 73T^{2} \) |
| 79 | \( 1 + 1.51T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 - 1.02T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−7.36487996755564136961921881380, −6.54505471319808728999741206450, −5.68811290585734011894948799852, −5.54705941807400003858110109858, −4.61632462518881978721758294739, −3.90696184206203487706691603123, −3.26591612235748056790844321251, −2.08154011956911503531257516595, −1.46024081843411772818239192786, 0,
1.46024081843411772818239192786, 2.08154011956911503531257516595, 3.26591612235748056790844321251, 3.90696184206203487706691603123, 4.61632462518881978721758294739, 5.54705941807400003858110109858, 5.68811290585734011894948799852, 6.54505471319808728999741206450, 7.36487996755564136961921881380