L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 5·11-s − 12-s − 3·13-s + 14-s + 16-s − 8·17-s − 18-s + 21-s + 5·22-s − 3·23-s + 24-s + 3·26-s − 27-s − 28-s + 4·29-s − 31-s − 32-s + 5·33-s + 8·34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.50·11-s − 0.288·12-s − 0.832·13-s + 0.267·14-s + 1/4·16-s − 1.94·17-s − 0.235·18-s + 0.218·21-s + 1.06·22-s − 0.625·23-s + 0.204·24-s + 0.588·26-s − 0.192·27-s − 0.188·28-s + 0.742·29-s − 0.179·31-s − 0.176·32-s + 0.870·33-s + 1.37·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 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
−12.77542065706900, −11.98971623276134, −11.92094288572539, −11.14663299912693, −10.94186113894030, −10.26036853117772, −10.16472646434309, −9.690755681995706, −9.056637544436496, −8.586878936452719, −8.213062537131451, −7.701657418427450, −7.064857761974271, −6.868725418348109, −6.355923360804815, −5.788118836779539, −5.246145113825162, −4.813073454155206, −4.380181128283998, −3.662802974711336, −2.792629665658096, −2.695218013987642, −1.875321306605628, −1.486102463492938, −0.2686846046400371, 0,
0.2686846046400371, 1.486102463492938, 1.875321306605628, 2.695218013987642, 2.792629665658096, 3.662802974711336, 4.380181128283998, 4.813073454155206, 5.246145113825162, 5.788118836779539, 6.355923360804815, 6.868725418348109, 7.064857761974271, 7.701657418427450, 8.213062537131451, 8.586878936452719, 9.056637544436496, 9.690755681995706, 10.16472646434309, 10.26036853117772, 10.94186113894030, 11.14663299912693, 11.92094288572539, 11.98971623276134, 12.77542065706900