L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 3·5-s − 2·6-s − 3·7-s + 9-s + 6·10-s − 5·11-s + 2·12-s + 2·13-s + 6·14-s − 3·15-s − 4·16-s − 6·17-s − 2·18-s − 6·19-s − 6·20-s − 3·21-s + 10·22-s + 9·23-s + 4·25-s − 4·26-s + 27-s − 6·28-s + 29-s + 6·30-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 1.34·5-s − 0.816·6-s − 1.13·7-s + 1/3·9-s + 1.89·10-s − 1.50·11-s + 0.577·12-s + 0.554·13-s + 1.60·14-s − 0.774·15-s − 16-s − 1.45·17-s − 0.471·18-s − 1.37·19-s − 1.34·20-s − 0.654·21-s + 2.13·22-s + 1.87·23-s + 4/5·25-s − 0.784·26-s + 0.192·27-s − 1.13·28-s + 0.185·29-s + 1.09·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 47 | \( 1 - T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−12.78597999000325082311656544421, −11.06455012201967038867455209089, −10.59810682607064930929411207650, −9.222291061463395227180870128738, −8.537254750564663506911107242219, −7.63249491006573329389585462503, −6.71008245044227369722426831675, −4.36466935124793744023969244712, −2.77908676833192943133298105927, 0,
2.77908676833192943133298105927, 4.36466935124793744023969244712, 6.71008245044227369722426831675, 7.63249491006573329389585462503, 8.537254750564663506911107242219, 9.222291061463395227180870128738, 10.59810682607064930929411207650, 11.06455012201967038867455209089, 12.78597999000325082311656544421