| L(s) = 1 | − 3-s − 5-s + 9-s + 13-s + 15-s − 6·17-s + 4·19-s + 25-s − 27-s − 2·29-s − 2·37-s − 39-s − 2·41-s − 4·43-s − 45-s + 4·47-s − 7·49-s + 6·51-s − 10·53-s − 4·57-s + 8·59-s − 2·61-s − 65-s − 4·67-s − 12·71-s − 6·73-s − 75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.277·13-s + 0.258·15-s − 1.45·17-s + 0.917·19-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.328·37-s − 0.160·39-s − 0.312·41-s − 0.609·43-s − 0.149·45-s + 0.583·47-s − 49-s + 0.840·51-s − 1.37·53-s − 0.529·57-s + 1.04·59-s − 0.256·61-s − 0.124·65-s − 0.488·67-s − 1.42·71-s − 0.702·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−9.029699286650580252378019897933, −8.255839595509746587661110158170, −7.30852958358231687995174425393, −6.66362668717235019653588903201, −5.75689880403616193455214552068, −4.84582010444282036922244924984, −4.05595867364853291162003882996, −2.96519757453610398171302719263, −1.55679555007052089516178041435, 0,
1.55679555007052089516178041435, 2.96519757453610398171302719263, 4.05595867364853291162003882996, 4.84582010444282036922244924984, 5.75689880403616193455214552068, 6.66362668717235019653588903201, 7.30852958358231687995174425393, 8.255839595509746587661110158170, 9.029699286650580252378019897933
