L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s + 9-s − 10-s + 3.41·11-s − 12-s + 15-s + 16-s + 1.41·17-s + 18-s − 2.82·19-s − 20-s + 3.41·22-s − 0.828·23-s − 24-s + 25-s − 27-s − 0.242·29-s + 30-s + 9.07·31-s + 32-s − 3.41·33-s + 1.41·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 1.02·11-s − 0.288·12-s + 0.258·15-s + 0.250·16-s + 0.342·17-s + 0.235·18-s − 0.648·19-s − 0.223·20-s + 0.727·22-s − 0.172·23-s − 0.204·24-s + 0.200·25-s − 0.192·27-s − 0.0450·29-s + 0.182·30-s + 1.62·31-s + 0.176·32-s − 0.594·33-s + 0.242·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.156868511\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.156868511\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 3.41T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 0.828T + 23T^{2} \) |
| 29 | \( 1 + 0.242T + 29T^{2} \) |
| 31 | \( 1 - 9.07T + 31T^{2} \) |
| 37 | \( 1 - 1.41T + 37T^{2} \) |
| 41 | \( 1 - 3.17T + 41T^{2} \) |
| 43 | \( 1 - 7.41T + 43T^{2} \) |
| 47 | \( 1 + 5.07T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 - 0.343T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 - 5.17T + 71T^{2} \) |
| 73 | \( 1 - 3.65T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + 3.17T + 97T^{2} \) |
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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
−9.637357838871249150992969140618, −8.627325750954760243245051288257, −7.75150497662837750195125132781, −6.80802711672635590355621544971, −6.25294328633980436113482368904, −5.33334734654136129989780436170, −4.34758789348115847590335941070, −3.78354034536825266062805513151, −2.48118261152276074185602676937, −1.02290488520011738099680178258,
1.02290488520011738099680178258, 2.48118261152276074185602676937, 3.78354034536825266062805513151, 4.34758789348115847590335941070, 5.33334734654136129989780436170, 6.25294328633980436113482368904, 6.80802711672635590355621544971, 7.75150497662837750195125132781, 8.627325750954760243245051288257, 9.637357838871249150992969140618