| L(s) = 1 | + 3·3-s + 20·7-s + 9·9-s − 16·11-s − 58·13-s − 38·17-s − 4·19-s + 60·21-s − 80·23-s + 27·27-s + 82·29-s + 8·31-s − 48·33-s − 426·37-s − 174·39-s − 246·41-s − 524·43-s − 464·47-s + 57·49-s − 114·51-s + 702·53-s − 12·57-s + 592·59-s + 574·61-s + 180·63-s − 172·67-s − 240·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.07·7-s + 1/3·9-s − 0.438·11-s − 1.23·13-s − 0.542·17-s − 0.0482·19-s + 0.623·21-s − 0.725·23-s + 0.192·27-s + 0.525·29-s + 0.0463·31-s − 0.253·33-s − 1.89·37-s − 0.714·39-s − 0.937·41-s − 1.85·43-s − 1.44·47-s + 0.166·49-s − 0.313·51-s + 1.81·53-s − 0.0278·57-s + 1.30·59-s + 1.20·61-s + 0.359·63-s − 0.313·67-s − 0.418·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 20 T + p^{3} T^{2} \) |
| 11 | \( 1 + 16 T + p^{3} T^{2} \) |
| 13 | \( 1 + 58 T + p^{3} T^{2} \) |
| 17 | \( 1 + 38 T + p^{3} T^{2} \) |
| 19 | \( 1 + 4 T + p^{3} T^{2} \) |
| 23 | \( 1 + 80 T + p^{3} T^{2} \) |
| 29 | \( 1 - 82 T + p^{3} T^{2} \) |
| 31 | \( 1 - 8 T + p^{3} T^{2} \) |
| 37 | \( 1 + 426 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 43 | \( 1 + 524 T + p^{3} T^{2} \) |
| 47 | \( 1 + 464 T + p^{3} T^{2} \) |
| 53 | \( 1 - 702 T + p^{3} T^{2} \) |
| 59 | \( 1 - 592 T + p^{3} T^{2} \) |
| 61 | \( 1 - 574 T + p^{3} T^{2} \) |
| 67 | \( 1 + 172 T + p^{3} T^{2} \) |
| 71 | \( 1 + 768 T + p^{3} T^{2} \) |
| 73 | \( 1 - 558 T + p^{3} T^{2} \) |
| 79 | \( 1 + 408 T + p^{3} T^{2} \) |
| 83 | \( 1 - 164 T + p^{3} T^{2} \) |
| 89 | \( 1 + 510 T + p^{3} T^{2} \) |
| 97 | \( 1 + 514 T + p^{3} T^{2} \) |
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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
−8.661866669224269565481907260401, −8.325757662671788554122809905613, −7.38975164334645866079613442341, −6.69880147011743094152562042439, −5.25368741174324151897871037653, −4.78028609485695993390311496027, −3.64134507814836246018594730620, −2.44279926683918990574622020367, −1.66931161354033887472717155188, 0,
1.66931161354033887472717155188, 2.44279926683918990574622020367, 3.64134507814836246018594730620, 4.78028609485695993390311496027, 5.25368741174324151897871037653, 6.69880147011743094152562042439, 7.38975164334645866079613442341, 8.325757662671788554122809905613, 8.661866669224269565481907260401
