L(s) = 1 | − 2·3-s − 7-s + 9-s − 5·11-s + 7·13-s + 7·19-s + 2·21-s + 23-s + 4·27-s + 5·29-s + 10·31-s + 10·33-s + 2·37-s − 14·39-s + 3·41-s − 9·43-s − 8·47-s − 6·49-s + 4·53-s − 14·57-s + 2·59-s − 6·61-s − 63-s − 8·67-s − 2·69-s + 2·71-s − 7·73-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.377·7-s + 1/3·9-s − 1.50·11-s + 1.94·13-s + 1.60·19-s + 0.436·21-s + 0.208·23-s + 0.769·27-s + 0.928·29-s + 1.79·31-s + 1.74·33-s + 0.328·37-s − 2.24·39-s + 0.468·41-s − 1.37·43-s − 1.16·47-s − 6/7·49-s + 0.549·53-s − 1.85·57-s + 0.260·59-s − 0.768·61-s − 0.125·63-s − 0.977·67-s − 0.240·69-s + 0.237·71-s − 0.819·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.169016240\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.169016240\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 17 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.76653352288099969145028474655, −6.79943956206547840931741677105, −6.23968339926991554049663136119, −5.73408216432204523856822174079, −5.07271423838019740012070687193, −4.47972411625611227529305533061, −3.24059389548277313460423595659, −2.90470772927929244453793300487, −1.39353018036701825169427986787, −0.60303334792834186976044852360,
0.60303334792834186976044852360, 1.39353018036701825169427986787, 2.90470772927929244453793300487, 3.24059389548277313460423595659, 4.47972411625611227529305533061, 5.07271423838019740012070687193, 5.73408216432204523856822174079, 6.23968339926991554049663136119, 6.79943956206547840931741677105, 7.76653352288099969145028474655