L(s) = 1 | + 1.71·3-s − 2.15·5-s − 0.0659·9-s − 3.02·11-s − 0.848·13-s − 3.68·15-s + 4.76·17-s + 3.90·19-s + 8.25·23-s − 0.372·25-s − 5.25·27-s + 0.0576·29-s − 4.30·31-s − 5.18·33-s + 0.475·37-s − 1.45·39-s − 41-s + 2.82·43-s + 0.141·45-s − 6.71·47-s + 8.15·51-s − 3.42·53-s + 6.51·55-s + 6.68·57-s − 5.72·59-s − 5.96·61-s + 1.82·65-s + ⋯ |
L(s) = 1 | + 0.988·3-s − 0.962·5-s − 0.0219·9-s − 0.912·11-s − 0.235·13-s − 0.951·15-s + 1.15·17-s + 0.894·19-s + 1.72·23-s − 0.0744·25-s − 1.01·27-s + 0.0107·29-s − 0.772·31-s − 0.902·33-s + 0.0781·37-s − 0.232·39-s − 0.156·41-s + 0.431·43-s + 0.0211·45-s − 0.980·47-s + 1.14·51-s − 0.471·53-s + 0.878·55-s + 0.885·57-s − 0.745·59-s − 0.764·61-s + 0.226·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 - 1.71T + 3T^{2} \) |
| 5 | \( 1 + 2.15T + 5T^{2} \) |
| 11 | \( 1 + 3.02T + 11T^{2} \) |
| 13 | \( 1 + 0.848T + 13T^{2} \) |
| 17 | \( 1 - 4.76T + 17T^{2} \) |
| 19 | \( 1 - 3.90T + 19T^{2} \) |
| 23 | \( 1 - 8.25T + 23T^{2} \) |
| 29 | \( 1 - 0.0576T + 29T^{2} \) |
| 31 | \( 1 + 4.30T + 31T^{2} \) |
| 37 | \( 1 - 0.475T + 37T^{2} \) |
| 43 | \( 1 - 2.82T + 43T^{2} \) |
| 47 | \( 1 + 6.71T + 47T^{2} \) |
| 53 | \( 1 + 3.42T + 53T^{2} \) |
| 59 | \( 1 + 5.72T + 59T^{2} \) |
| 61 | \( 1 + 5.96T + 61T^{2} \) |
| 67 | \( 1 + 9.53T + 67T^{2} \) |
| 71 | \( 1 + 1.84T + 71T^{2} \) |
| 73 | \( 1 + 3.89T + 73T^{2} \) |
| 79 | \( 1 - 5.11T + 79T^{2} \) |
| 83 | \( 1 - 7.41T + 83T^{2} \) |
| 89 | \( 1 - 0.305T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−7.58763626506309092459767897481, −7.25671361779976541339371387409, −6.05640789003655280298359225429, −5.25721687900625079451535282025, −4.67452805743869483873551752359, −3.51156791957912783191584415111, −3.25439597505372735466218580004, −2.49965970971102682397102561749, −1.29350661697533612052774485329, 0,
1.29350661697533612052774485329, 2.49965970971102682397102561749, 3.25439597505372735466218580004, 3.51156791957912783191584415111, 4.67452805743869483873551752359, 5.25721687900625079451535282025, 6.05640789003655280298359225429, 7.25671361779976541339371387409, 7.58763626506309092459767897481