| L(s) = 1 | − 11·7-s − 23·13-s − 37·19-s + 25·25-s + 46·31-s + 73·37-s − 22·43-s + 72·49-s − 47·61-s − 13·67-s + 143·73-s − 11·79-s + 253·91-s − 169·97-s + 157·103-s + 214·109-s + ⋯ |
| L(s) = 1 | − 1.57·7-s − 1.76·13-s − 1.94·19-s + 25-s + 1.48·31-s + 1.97·37-s − 0.511·43-s + 1.46·49-s − 0.770·61-s − 0.194·67-s + 1.95·73-s − 0.139·79-s + 2.78·91-s − 1.74·97-s + 1.52·103-s + 1.96·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9345057752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9345057752\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - p T )( 1 + p T ) \) |
| 7 | \( 1 + 11 T + p^{2} T^{2} \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 + 23 T + p^{2} T^{2} \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( 1 + 37 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 - 46 T + p^{2} T^{2} \) |
| 37 | \( 1 - 73 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 + 22 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 + 47 T + p^{2} T^{2} \) |
| 67 | \( 1 + 13 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 - 143 T + p^{2} T^{2} \) |
| 79 | \( 1 + 11 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 + 169 T + p^{2} T^{2} \) |
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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
−9.279763720288012814431993250947, −8.403641712623959853545594974365, −7.46101494986081213231261223752, −6.57896858909440005116418770661, −6.21391593609302191926128632717, −4.92318939010656033368044513128, −4.18745093350585495477722553864, −2.96312659714924292841657889967, −2.35514317189522414761135866902, −0.49954522553053975745240448576,
0.49954522553053975745240448576, 2.35514317189522414761135866902, 2.96312659714924292841657889967, 4.18745093350585495477722553864, 4.92318939010656033368044513128, 6.21391593609302191926128632717, 6.57896858909440005116418770661, 7.46101494986081213231261223752, 8.403641712623959853545594974365, 9.279763720288012814431993250947
