| L(s) = 1 | + 4·5-s − 4·11-s − 13-s − 4·17-s + 19-s − 4·23-s + 11·25-s + 4·31-s − 9·37-s − 8·43-s − 12·47-s + 8·53-s − 16·55-s + 4·59-s + 5·61-s − 4·65-s + 11·67-s − 8·71-s − 73-s − 5·79-s + 8·83-s − 16·85-s + 12·89-s + 4·95-s − 5·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.20·11-s − 0.277·13-s − 0.970·17-s + 0.229·19-s − 0.834·23-s + 11/5·25-s + 0.718·31-s − 1.47·37-s − 1.21·43-s − 1.75·47-s + 1.09·53-s − 2.15·55-s + 0.520·59-s + 0.640·61-s − 0.496·65-s + 1.34·67-s − 0.949·71-s − 0.117·73-s − 0.562·79-s + 0.878·83-s − 1.73·85-s + 1.27·89-s + 0.410·95-s − 0.507·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−17.01807686053691, −16.15592719679277, −15.89577560579056, −15.01045248718039, −14.53036195262651, −13.80961936370978, −13.32095213178199, −13.17938373201509, −12.32403531851714, −11.65741033605808, −10.82180948927141, −10.24849833456888, −9.960858307681918, −9.322829877860240, −8.558613764687297, −8.092997792439921, −7.048746518564329, −6.570649661807295, −5.904994013053832, −5.137216519163650, −4.931199989921488, −3.697633195400231, −2.669896353127453, −2.228073589052293, −1.441782079207978, 0,
1.441782079207978, 2.228073589052293, 2.669896353127453, 3.697633195400231, 4.931199989921488, 5.137216519163650, 5.904994013053832, 6.570649661807295, 7.048746518564329, 8.092997792439921, 8.558613764687297, 9.322829877860240, 9.960858307681918, 10.24849833456888, 10.82180948927141, 11.65741033605808, 12.32403531851714, 13.17938373201509, 13.32095213178199, 13.80961936370978, 14.53036195262651, 15.01045248718039, 15.89577560579056, 16.15592719679277, 17.01807686053691
