L(s) = 1 | + 7·13-s + 19-s − 5·25-s + 4·31-s − 37-s + 8·43-s + 13·61-s + 11·67-s − 17·73-s − 13·79-s − 5·97-s + 7·103-s + 2·109-s + ⋯ |
L(s) = 1 | + 1.94·13-s + 0.229·19-s − 25-s + 0.718·31-s − 0.164·37-s + 1.21·43-s + 1.66·61-s + 1.34·67-s − 1.98·73-s − 1.46·79-s − 0.507·97-s + 0.689·103-s + 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.185717102\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.185717102\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 17 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−8.321813700481147925892685842672, −7.51552880521900195309523002532, −6.70078665608916533782840710374, −5.98139804295706451305092616431, −5.49030247404272458638604331312, −4.33573970367631074650890566231, −3.77813458199186956523245005153, −2.91173766719274595341337420733, −1.79680621722010121991203363355, −0.833659174677608845252044473607,
0.833659174677608845252044473607, 1.79680621722010121991203363355, 2.91173766719274595341337420733, 3.77813458199186956523245005153, 4.33573970367631074650890566231, 5.49030247404272458638604331312, 5.98139804295706451305092616431, 6.70078665608916533782840710374, 7.51552880521900195309523002532, 8.321813700481147925892685842672