| L(s) = 1 | − 3-s + 9-s + 2·11-s + 13-s − 3·19-s − 27-s − 9·31-s − 2·33-s + 3·37-s − 39-s + 2·41-s + 3·43-s + 6·47-s + 3·57-s + 4·59-s − 2·61-s + 5·67-s + 14·71-s − 73-s + 9·79-s + 81-s − 6·83-s − 4·89-s + 9·93-s + 14·97-s + 2·99-s + 101-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.603·11-s + 0.277·13-s − 0.688·19-s − 0.192·27-s − 1.61·31-s − 0.348·33-s + 0.493·37-s − 0.160·39-s + 0.312·41-s + 0.457·43-s + 0.875·47-s + 0.397·57-s + 0.520·59-s − 0.256·61-s + 0.610·67-s + 1.66·71-s − 0.117·73-s + 1.01·79-s + 1/9·81-s − 0.658·83-s − 0.423·89-s + 0.933·93-s + 1.42·97-s + 0.201·99-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.08856740958357, −12.66404793210337, −12.14029411704811, −11.84641949241366, −11.17220622771598, −10.79523140616766, −10.65144517951262, −9.780654738209112, −9.451438102718324, −9.030972145475529, −8.431102266872363, −7.991043854038467, −7.323544711458383, −6.989893696285204, −6.376479299939824, −6.039777690678569, −5.434526817675010, −5.048020434033692, −4.296878097580145, −3.914197208786037, −3.489364531521449, −2.586113307588891, −2.097227719074964, −1.390657172013036, −0.7849479947612411, 0,
0.7849479947612411, 1.390657172013036, 2.097227719074964, 2.586113307588891, 3.489364531521449, 3.914197208786037, 4.296878097580145, 5.048020434033692, 5.434526817675010, 6.039777690678569, 6.376479299939824, 6.989893696285204, 7.323544711458383, 7.991043854038467, 8.431102266872363, 9.030972145475529, 9.451438102718324, 9.780654738209112, 10.65144517951262, 10.79523140616766, 11.17220622771598, 11.84641949241366, 12.14029411704811, 12.66404793210337, 13.08856740958357
