L(s) = 1 | + 3-s + 5-s + 7-s + 9-s − 3·11-s − 13-s + 15-s − 5·17-s − 2·19-s + 21-s − 3·23-s + 25-s + 27-s − 4·31-s − 3·33-s + 35-s + 37-s − 39-s + 9·41-s + 2·43-s + 45-s + 8·47-s − 6·49-s − 5·51-s − 53-s − 3·55-s − 2·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.277·13-s + 0.258·15-s − 1.21·17-s − 0.458·19-s + 0.218·21-s − 0.625·23-s + 1/5·25-s + 0.192·27-s − 0.718·31-s − 0.522·33-s + 0.169·35-s + 0.164·37-s − 0.160·39-s + 1.40·41-s + 0.304·43-s + 0.149·45-s + 1.16·47-s − 6/7·49-s − 0.700·51-s − 0.137·53-s − 0.404·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−7.51267103666363647636595018130, −7.31918358398006824210797765877, −6.15834138196877387997637643255, −5.66666054720517678180962319847, −4.60066996685305628812936351416, −4.22336140015269997485244311013, −2.96571908368185861789583145156, −2.36635639623477881054823011237, −1.57061372831709803750701887248, 0,
1.57061372831709803750701887248, 2.36635639623477881054823011237, 2.96571908368185861789583145156, 4.22336140015269997485244311013, 4.60066996685305628812936351416, 5.66666054720517678180962319847, 6.15834138196877387997637643255, 7.31918358398006824210797765877, 7.51267103666363647636595018130