| L(s) = 1 | + 3-s + 3·5-s − 2·9-s + 4·11-s + 5·13-s + 3·15-s − 5·17-s + 23-s + 4·25-s − 5·27-s − 3·29-s + 4·31-s + 4·33-s − 2·37-s + 5·39-s + 5·41-s + 11·43-s − 6·45-s + 5·47-s − 7·49-s − 5·51-s + 9·53-s + 12·55-s + 13·59-s − 61-s + 15·65-s − 5·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s − 2/3·9-s + 1.20·11-s + 1.38·13-s + 0.774·15-s − 1.21·17-s + 0.208·23-s + 4/5·25-s − 0.962·27-s − 0.557·29-s + 0.718·31-s + 0.696·33-s − 0.328·37-s + 0.800·39-s + 0.780·41-s + 1.67·43-s − 0.894·45-s + 0.729·47-s − 49-s − 0.700·51-s + 1.23·53-s + 1.61·55-s + 1.69·59-s − 0.128·61-s + 1.86·65-s − 0.610·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.533063094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.533063094\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−8.423878990021024655213982027767, −7.35652637536519506079429545238, −6.47767067267187089609187392355, −6.06720825087535552854122180257, −5.46823062601523206258085766163, −4.29871619218814968358130057634, −3.65835659796154226528357515686, −2.63511910399759571063777343605, −1.97626011047699633972964765499, −1.02319488702079724960627868609,
1.02319488702079724960627868609, 1.97626011047699633972964765499, 2.63511910399759571063777343605, 3.65835659796154226528357515686, 4.29871619218814968358130057634, 5.46823062601523206258085766163, 6.06720825087535552854122180257, 6.47767067267187089609187392355, 7.35652637536519506079429545238, 8.423878990021024655213982027767
