| L(s) = 1 | − 13·7-s + 13-s − 11·19-s + 25·25-s − 46·31-s − 47·37-s + 22·43-s + 120·49-s + 121·61-s + 109·67-s − 97·73-s + 131·79-s − 13·91-s + 167·97-s − 37·103-s + 214·109-s + ⋯ |
| L(s) = 1 | − 1.85·7-s + 1/13·13-s − 0.578·19-s + 25-s − 1.48·31-s − 1.27·37-s + 0.511·43-s + 2.44·49-s + 1.98·61-s + 1.62·67-s − 1.32·73-s + 1.65·79-s − 1/7·91-s + 1.72·97-s − 0.359·103-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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.134380994\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.134380994\) |
| \(L(2)\) |
|
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 | \( 1 + 13 T + p^{2} T^{2} \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 - T + p^{2} T^{2} \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( 1 + 11 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 + 46 T + p^{2} T^{2} \) |
| 37 | \( 1 + 47 T + p^{2} T^{2} \) |
| 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 | \( 1 - 121 T + p^{2} T^{2} \) |
| 67 | \( 1 - 109 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 + 97 T + p^{2} T^{2} \) |
| 79 | \( 1 - 131 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 - 167 T + p^{2} T^{2} \) |
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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
−9.123801051846413103330406610828, −8.581542413894096776602464967478, −7.31632803758994342277649741183, −6.75334856964734995500111286248, −6.03350481902915107763072277818, −5.14416190105392918223198545584, −3.87558196289165821760800651259, −3.26319864202481880782879530014, −2.19964461885454952227107203126, −0.56120774588241737555213711845,
0.56120774588241737555213711845, 2.19964461885454952227107203126, 3.26319864202481880782879530014, 3.87558196289165821760800651259, 5.14416190105392918223198545584, 6.03350481902915107763072277818, 6.75334856964734995500111286248, 7.31632803758994342277649741183, 8.581542413894096776602464967478, 9.123801051846413103330406610828
