L(s) = 1 | − 2-s + 4-s − 5-s − 4·7-s − 8-s + 10-s − 11-s − 4·13-s + 4·14-s + 16-s − 17-s − 2·19-s − 20-s + 22-s − 2·23-s + 25-s + 4·26-s − 4·28-s − 2·29-s − 32-s + 34-s + 4·35-s + 10·37-s + 2·38-s + 40-s − 10·41-s + 2·43-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s − 0.353·8-s + 0.316·10-s − 0.301·11-s − 1.10·13-s + 1.06·14-s + 1/4·16-s − 0.242·17-s − 0.458·19-s − 0.223·20-s + 0.213·22-s − 0.417·23-s + 1/5·25-s + 0.784·26-s − 0.755·28-s − 0.371·29-s − 0.176·32-s + 0.171·34-s + 0.676·35-s + 1.64·37-s + 0.324·38-s + 0.158·40-s − 1.56·41-s + 0.304·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−16.34278730207397, −15.66883834943621, −15.11263376251177, −14.89755780492055, −13.83102854531416, −13.40274956659460, −12.60042363889376, −12.38167275609501, −11.74089944900311, −11.02475917233029, −10.41361577190852, −9.890220305788905, −9.477723638319432, −8.865403737920613, −8.192879444911340, −7.509940430855706, −7.045360723859248, −6.435050272844027, −5.841507549348028, −5.023219922131779, −4.125687554391892, −3.494623836529722, −2.651171605854609, −2.185597918146964, −0.7481226000441284, 0,
0.7481226000441284, 2.185597918146964, 2.651171605854609, 3.494623836529722, 4.125687554391892, 5.023219922131779, 5.841507549348028, 6.435050272844027, 7.045360723859248, 7.509940430855706, 8.192879444911340, 8.865403737920613, 9.477723638319432, 9.890220305788905, 10.41361577190852, 11.02475917233029, 11.74089944900311, 12.38167275609501, 12.60042363889376, 13.40274956659460, 13.83102854531416, 14.89755780492055, 15.11263376251177, 15.66883834943621, 16.34278730207397