L(s) = 1 | + 2-s − 4-s − 4·5-s − 4·7-s − 3·8-s − 3·9-s − 4·10-s + 4·11-s − 4·14-s − 16-s + 2·17-s − 3·18-s − 4·19-s + 4·20-s + 4·22-s + 4·23-s + 11·25-s + 4·28-s − 4·29-s − 2·31-s + 5·32-s + 2·34-s + 16·35-s + 3·36-s + 6·37-s − 4·38-s + 12·40-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.78·5-s − 1.51·7-s − 1.06·8-s − 9-s − 1.26·10-s + 1.20·11-s − 1.06·14-s − 1/4·16-s + 0.485·17-s − 0.707·18-s − 0.917·19-s + 0.894·20-s + 0.852·22-s + 0.834·23-s + 11/5·25-s + 0.755·28-s − 0.742·29-s − 0.359·31-s + 0.883·32-s + 0.342·34-s + 2.70·35-s + 1/2·36-s + 0.986·37-s − 0.648·38-s + 1.89·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 73 | \( 1 - T \) |
| 137 | \( 1 - T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−16.83818430649483, −16.36596747611176, −15.60478010452076, −15.15281101720038, −14.57421708887065, −14.28379845623159, −13.30802846135986, −12.86169562443263, −12.24172147358369, −12.01212029107784, −11.19099123788464, −10.81295475969948, −9.563796200572135, −9.252289626241779, −8.647133885305995, −8.019494782556084, −7.221127011782987, −6.479878083322923, −6.042081009666289, −5.154411930752986, −4.254429892912670, −3.824786579941877, −3.309544997528499, −2.720165075912424, −0.7738720728652220, 0,
0.7738720728652220, 2.720165075912424, 3.309544997528499, 3.824786579941877, 4.254429892912670, 5.154411930752986, 6.042081009666289, 6.479878083322923, 7.221127011782987, 8.019494782556084, 8.647133885305995, 9.252289626241779, 9.563796200572135, 10.81295475969948, 11.19099123788464, 12.01212029107784, 12.24172147358369, 12.86169562443263, 13.30802846135986, 14.28379845623159, 14.57421708887065, 15.15281101720038, 15.60478010452076, 16.36596747611176, 16.83818430649483