| L(s) = 1 | + 2·3-s − 2·5-s + 9-s − 11-s + 4·13-s − 4·15-s + 4·19-s − 4·23-s − 25-s − 4·27-s + 2·29-s − 10·31-s − 2·33-s − 6·37-s + 8·39-s + 4·43-s − 2·45-s + 10·47-s − 14·53-s + 2·55-s + 8·57-s + 10·59-s + 8·61-s − 8·65-s − 8·67-s − 8·69-s + 4·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 1/3·9-s − 0.301·11-s + 1.10·13-s − 1.03·15-s + 0.917·19-s − 0.834·23-s − 1/5·25-s − 0.769·27-s + 0.371·29-s − 1.79·31-s − 0.348·33-s − 0.986·37-s + 1.28·39-s + 0.609·43-s − 0.298·45-s + 1.45·47-s − 1.92·53-s + 0.269·55-s + 1.05·57-s + 1.30·59-s + 1.02·61-s − 0.992·65-s − 0.977·67-s − 0.963·69-s + 0.474·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.46164400989651408349485864098, −7.14402834064955129249452978674, −5.95282824378351229072848673006, −5.44276931191501819834548274842, −4.28458254538175390142522268801, −3.70692833339407757257172884768, −3.24853321699752513715801783640, −2.33092974468556821917785481952, −1.39134982267066587201348228305, 0,
1.39134982267066587201348228305, 2.33092974468556821917785481952, 3.24853321699752513715801783640, 3.70692833339407757257172884768, 4.28458254538175390142522268801, 5.44276931191501819834548274842, 5.95282824378351229072848673006, 7.14402834064955129249452978674, 7.46164400989651408349485864098
