| L(s) = 1 | − 2-s − 2.79·3-s + 4-s + 0.116·5-s + 2.79·6-s + 2.35·7-s − 8-s + 4.83·9-s − 0.116·10-s − 2.79·12-s + 2.88·13-s − 2.35·14-s − 0.324·15-s + 16-s + 2.91·17-s − 4.83·18-s + 19-s + 0.116·20-s − 6.59·21-s − 1.44·23-s + 2.79·24-s − 4.98·25-s − 2.88·26-s − 5.12·27-s + 2.35·28-s + 5.15·29-s + 0.324·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.61·3-s + 0.5·4-s + 0.0519·5-s + 1.14·6-s + 0.890·7-s − 0.353·8-s + 1.61·9-s − 0.0367·10-s − 0.807·12-s + 0.799·13-s − 0.630·14-s − 0.0838·15-s + 0.250·16-s + 0.706·17-s − 1.13·18-s + 0.229·19-s + 0.0259·20-s − 1.43·21-s − 0.300·23-s + 0.571·24-s − 0.997·25-s − 0.565·26-s − 0.985·27-s + 0.445·28-s + 0.957·29-s + 0.0593·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9944892515\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9944892515\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.79T + 3T^{2} \) |
| 5 | \( 1 - 0.116T + 5T^{2} \) |
| 7 | \( 1 - 2.35T + 7T^{2} \) |
| 13 | \( 1 - 2.88T + 13T^{2} \) |
| 17 | \( 1 - 2.91T + 17T^{2} \) |
| 23 | \( 1 + 1.44T + 23T^{2} \) |
| 29 | \( 1 - 5.15T + 29T^{2} \) |
| 31 | \( 1 - 8.32T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 - 9.98T + 41T^{2} \) |
| 43 | \( 1 + 2.76T + 43T^{2} \) |
| 47 | \( 1 - 4.23T + 47T^{2} \) |
| 53 | \( 1 - 5.56T + 53T^{2} \) |
| 59 | \( 1 + 8.51T + 59T^{2} \) |
| 61 | \( 1 - 2.71T + 61T^{2} \) |
| 67 | \( 1 - 3.41T + 67T^{2} \) |
| 71 | \( 1 + 1.16T + 71T^{2} \) |
| 73 | \( 1 + 1.31T + 73T^{2} \) |
| 79 | \( 1 + 1.93T + 79T^{2} \) |
| 83 | \( 1 - 4.05T + 83T^{2} \) |
| 89 | \( 1 + 9.25T + 89T^{2} \) |
| 97 | \( 1 - 8.71T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.053591748191232133555209651135, −7.78372585118013076089623522044, −6.73632707609822306936930857545, −6.08273216385337205320955981929, −5.64153436616335667169365166609, −4.72990298617147611073059594334, −4.04779848926035873181339456592, −2.65341685761541184894731410214, −1.38484509504537620049637323623, −0.76812334043451348493565801371,
0.76812334043451348493565801371, 1.38484509504537620049637323623, 2.65341685761541184894731410214, 4.04779848926035873181339456592, 4.72990298617147611073059594334, 5.64153436616335667169365166609, 6.08273216385337205320955981929, 6.73632707609822306936930857545, 7.78372585118013076089623522044, 8.053591748191232133555209651135
