| L(s) = 1 | − 0.772·3-s + 2.52·5-s − 1.43·7-s − 2.40·9-s − 2.28·11-s − 0.693·13-s − 1.95·15-s − 17-s − 7.02·19-s + 1.11·21-s − 3.49·23-s + 1.37·25-s + 4.17·27-s − 3.75·29-s + 2.60·31-s + 1.76·33-s − 3.62·35-s + 10.7·37-s + 0.535·39-s − 2.26·41-s + 0.738·43-s − 6.06·45-s − 7.84·47-s − 4.93·49-s + 0.772·51-s − 4.53·53-s − 5.76·55-s + ⋯ |
| L(s) = 1 | − 0.445·3-s + 1.12·5-s − 0.543·7-s − 0.801·9-s − 0.688·11-s − 0.192·13-s − 0.503·15-s − 0.242·17-s − 1.61·19-s + 0.242·21-s − 0.729·23-s + 0.275·25-s + 0.803·27-s − 0.696·29-s + 0.467·31-s + 0.307·33-s − 0.613·35-s + 1.76·37-s + 0.0857·39-s − 0.353·41-s + 0.112·43-s − 0.904·45-s − 1.14·47-s − 0.704·49-s + 0.108·51-s − 0.623·53-s − 0.777·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.094690155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.094690155\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 0.772T + 3T^{2} \) |
| 5 | \( 1 - 2.52T + 5T^{2} \) |
| 7 | \( 1 + 1.43T + 7T^{2} \) |
| 11 | \( 1 + 2.28T + 11T^{2} \) |
| 13 | \( 1 + 0.693T + 13T^{2} \) |
| 19 | \( 1 + 7.02T + 19T^{2} \) |
| 23 | \( 1 + 3.49T + 23T^{2} \) |
| 29 | \( 1 + 3.75T + 29T^{2} \) |
| 31 | \( 1 - 2.60T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 + 2.26T + 41T^{2} \) |
| 43 | \( 1 - 0.738T + 43T^{2} \) |
| 47 | \( 1 + 7.84T + 47T^{2} \) |
| 53 | \( 1 + 4.53T + 53T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 + 0.228T + 79T^{2} \) |
| 83 | \( 1 + 17.4T + 83T^{2} \) |
| 89 | \( 1 + 8.61T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−8.056971800201066891984712273229, −6.83284328132415981319147149607, −6.32210805288258946019850139743, −5.86331968951488022540535245385, −5.20432879007498712754371600725, −4.43996801329111422040734733210, −3.42168767243986291187066541563, −2.44363107630013041232995393973, −2.02582967708641159729397938466, −0.49560147157144470045093685199,
0.49560147157144470045093685199, 2.02582967708641159729397938466, 2.44363107630013041232995393973, 3.42168767243986291187066541563, 4.43996801329111422040734733210, 5.20432879007498712754371600725, 5.86331968951488022540535245385, 6.32210805288258946019850139743, 6.83284328132415981319147149607, 8.056971800201066891984712273229
