| L(s) = 1 | + 3·5-s − 7-s + 3·11-s + 2·19-s − 6·23-s + 4·25-s − 6·29-s + 5·31-s − 3·35-s − 2·37-s − 6·41-s + 10·43-s − 6·47-s − 6·49-s − 9·53-s + 9·55-s − 12·59-s + 8·61-s + 14·67-s + 7·73-s − 3·77-s − 8·79-s + 3·83-s − 18·89-s + 6·95-s + 97-s + 101-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.377·7-s + 0.904·11-s + 0.458·19-s − 1.25·23-s + 4/5·25-s − 1.11·29-s + 0.898·31-s − 0.507·35-s − 0.328·37-s − 0.937·41-s + 1.52·43-s − 0.875·47-s − 6/7·49-s − 1.23·53-s + 1.21·55-s − 1.56·59-s + 1.02·61-s + 1.71·67-s + 0.819·73-s − 0.341·77-s − 0.900·79-s + 0.329·83-s − 1.90·89-s + 0.615·95-s + 0.101·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−14.28954089795022, −13.91991886366624, −13.42343963491726, −12.84914284111857, −12.47419208332951, −11.86261066656611, −11.31863042387061, −10.85686852714670, −10.02798332910039, −9.820742869797205, −9.434380926027266, −8.928854291033229, −8.221178202665519, −7.750570007849616, −6.914841174455778, −6.536416652811248, −6.009238826316853, −5.628424791101710, −4.955771552946744, −4.291439402890063, −3.606703429353559, −3.059953779876800, −2.207671357253977, −1.758413086499338, −1.090796286878279, 0,
1.090796286878279, 1.758413086499338, 2.207671357253977, 3.059953779876800, 3.606703429353559, 4.291439402890063, 4.955771552946744, 5.628424791101710, 6.009238826316853, 6.536416652811248, 6.914841174455778, 7.750570007849616, 8.221178202665519, 8.928854291033229, 9.434380926027266, 9.820742869797205, 10.02798332910039, 10.85686852714670, 11.31863042387061, 11.86261066656611, 12.47419208332951, 12.84914284111857, 13.42343963491726, 13.91991886366624, 14.28954089795022
