L(s) = 1 | − 2-s + 2.53·3-s + 4-s − 1.21·5-s − 2.53·6-s − 8-s + 3.42·9-s + 1.21·10-s + 3.74·11-s + 2.53·12-s − 2.95·13-s − 3.06·15-s + 16-s − 6.53·17-s − 3.42·18-s − 4.95·19-s − 1.21·20-s − 3.74·22-s + 6.02·23-s − 2.53·24-s − 3.53·25-s + 2.95·26-s + 1.06·27-s − 0.901·29-s + 3.06·30-s − 9.48·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.46·3-s + 0.5·4-s − 0.541·5-s − 1.03·6-s − 0.353·8-s + 1.14·9-s + 0.382·10-s + 1.12·11-s + 0.731·12-s − 0.819·13-s − 0.792·15-s + 0.250·16-s − 1.58·17-s − 0.806·18-s − 1.13·19-s − 0.270·20-s − 0.798·22-s + 1.25·23-s − 0.517·24-s − 0.706·25-s + 0.579·26-s + 0.205·27-s − 0.167·29-s + 0.560·30-s − 1.70·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 3 | \( 1 - 2.53T + 3T^{2} \) |
| 5 | \( 1 + 1.21T + 5T^{2} \) |
| 11 | \( 1 - 3.74T + 11T^{2} \) |
| 13 | \( 1 + 2.95T + 13T^{2} \) |
| 17 | \( 1 + 6.53T + 17T^{2} \) |
| 19 | \( 1 + 4.95T + 19T^{2} \) |
| 23 | \( 1 - 6.02T + 23T^{2} \) |
| 29 | \( 1 + 0.901T + 29T^{2} \) |
| 31 | \( 1 + 9.48T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 43 | \( 1 + 9.06T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 5.32T + 53T^{2} \) |
| 59 | \( 1 + 2.36T + 59T^{2} \) |
| 61 | \( 1 + 3.76T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 - 1.06T + 73T^{2} \) |
| 79 | \( 1 - 4.84T + 79T^{2} \) |
| 83 | \( 1 - 1.29T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 + 4.13T + 97T^{2} \) |
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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
−8.241897343057805096643253183059, −7.53619100298152590604165517764, −6.90150674072391108696303836394, −6.25203328286861974565347052523, −4.78472896555252136205883272798, −4.04339558848267215510382112909, −3.29180279926585272497948266651, −2.34459212189848846270867947138, −1.67824977931410804250975117303, 0,
1.67824977931410804250975117303, 2.34459212189848846270867947138, 3.29180279926585272497948266651, 4.04339558848267215510382112909, 4.78472896555252136205883272798, 6.25203328286861974565347052523, 6.90150674072391108696303836394, 7.53619100298152590604165517764, 8.241897343057805096643253183059