| L(s) = 1 | + 2·3-s + 4·7-s + 9-s − 11-s + 6·13-s − 2·17-s − 4·19-s + 8·21-s + 6·23-s − 4·27-s − 2·29-s − 8·31-s − 2·33-s + 8·37-s + 12·39-s + 6·41-s + 12·43-s + 10·47-s + 9·49-s − 4·51-s − 8·57-s + 4·59-s − 10·61-s + 4·63-s + 2·67-s + 12·69-s + 8·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.51·7-s + 1/3·9-s − 0.301·11-s + 1.66·13-s − 0.485·17-s − 0.917·19-s + 1.74·21-s + 1.25·23-s − 0.769·27-s − 0.371·29-s − 1.43·31-s − 0.348·33-s + 1.31·37-s + 1.92·39-s + 0.937·41-s + 1.82·43-s + 1.45·47-s + 9/7·49-s − 0.560·51-s − 1.05·57-s + 0.520·59-s − 1.28·61-s + 0.503·63-s + 0.244·67-s + 1.44·69-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.743964678\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.743964678\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−8.428952625359810959381045717968, −7.78810547280425081611087515818, −7.23882664967131620094169749537, −6.06722921790833825540180903158, −5.44255287090529663970265958380, −4.33283846251475049114934623641, −3.90273373929304638709386348539, −2.77795406152189934158119772328, −2.05680716214755609125321225100, −1.11717290376769012198383328813,
1.11717290376769012198383328813, 2.05680716214755609125321225100, 2.77795406152189934158119772328, 3.90273373929304638709386348539, 4.33283846251475049114934623641, 5.44255287090529663970265958380, 6.06722921790833825540180903158, 7.23882664967131620094169749537, 7.78810547280425081611087515818, 8.428952625359810959381045717968
