| L(s) = 1 | + 1.28·2-s + 3-s − 0.361·4-s + 1.28·6-s − 3.00·7-s − 3.02·8-s + 9-s + 2.74·11-s − 0.361·12-s + 1.26·13-s − 3.84·14-s − 3.14·16-s + 0.0436·17-s + 1.28·18-s + 0.625·19-s − 3.00·21-s + 3.51·22-s + 5.48·23-s − 3.02·24-s + 1.61·26-s + 27-s + 1.08·28-s − 3.98·29-s − 1.30·31-s + 2.01·32-s + 2.74·33-s + 0.0558·34-s + ⋯ |
| L(s) = 1 | + 0.905·2-s + 0.577·3-s − 0.180·4-s + 0.522·6-s − 1.13·7-s − 1.06·8-s + 0.333·9-s + 0.828·11-s − 0.104·12-s + 0.349·13-s − 1.02·14-s − 0.786·16-s + 0.0105·17-s + 0.301·18-s + 0.143·19-s − 0.654·21-s + 0.749·22-s + 1.14·23-s − 0.617·24-s + 0.316·26-s + 0.192·27-s + 0.204·28-s − 0.740·29-s − 0.233·31-s + 0.356·32-s + 0.478·33-s + 0.00958·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.852515838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.852515838\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - 1.28T + 2T^{2} \) |
| 7 | \( 1 + 3.00T + 7T^{2} \) |
| 11 | \( 1 - 2.74T + 11T^{2} \) |
| 13 | \( 1 - 1.26T + 13T^{2} \) |
| 17 | \( 1 - 0.0436T + 17T^{2} \) |
| 19 | \( 1 - 0.625T + 19T^{2} \) |
| 23 | \( 1 - 5.48T + 23T^{2} \) |
| 29 | \( 1 + 3.98T + 29T^{2} \) |
| 31 | \( 1 + 1.30T + 31T^{2} \) |
| 37 | \( 1 - 5.54T + 37T^{2} \) |
| 41 | \( 1 - 6.26T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 53 | \( 1 - 5.93T + 53T^{2} \) |
| 59 | \( 1 + 2.51T + 59T^{2} \) |
| 61 | \( 1 - 0.757T + 61T^{2} \) |
| 67 | \( 1 - 1.99T + 67T^{2} \) |
| 71 | \( 1 + 2.84T + 71T^{2} \) |
| 73 | \( 1 - 4.79T + 73T^{2} \) |
| 79 | \( 1 - 7.03T + 79T^{2} \) |
| 83 | \( 1 - 6.52T + 83T^{2} \) |
| 89 | \( 1 - 2.62T + 89T^{2} \) |
| 97 | \( 1 - 1.36T + 97T^{2} \) |
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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
−8.832104582223521389046976632507, −7.76039541796696502712389289173, −6.90950235148546683456172389300, −6.22505245262118159290125120180, −5.59025510601319400952951477761, −4.54392552687302007799965526198, −3.83211159072856256329534762526, −3.27279099953424647460945985262, −2.43762841124659760990395465805, −0.851967587765644656240725072588,
0.851967587765644656240725072588, 2.43762841124659760990395465805, 3.27279099953424647460945985262, 3.83211159072856256329534762526, 4.54392552687302007799965526198, 5.59025510601319400952951477761, 6.22505245262118159290125120180, 6.90950235148546683456172389300, 7.76039541796696502712389289173, 8.832104582223521389046976632507
