| L(s) = 1 | − 0.604·3-s + 2.44·5-s + 2.35·7-s − 2.63·9-s + 0.468·11-s + 3.50·13-s − 1.48·15-s − 6.18·17-s + 19-s − 1.42·21-s − 7.62·23-s + 1.00·25-s + 3.40·27-s + 1.30·29-s − 1.64·31-s − 0.283·33-s + 5.76·35-s − 9.36·37-s − 2.12·39-s − 0.982·41-s − 10.0·43-s − 6.45·45-s − 2.43·47-s − 1.45·49-s + 3.73·51-s + 2.13·53-s + 1.14·55-s + ⋯ |
| L(s) = 1 | − 0.349·3-s + 1.09·5-s + 0.889·7-s − 0.877·9-s + 0.141·11-s + 0.973·13-s − 0.382·15-s − 1.49·17-s + 0.229·19-s − 0.310·21-s − 1.58·23-s + 0.200·25-s + 0.655·27-s + 0.241·29-s − 0.295·31-s − 0.0493·33-s + 0.974·35-s − 1.53·37-s − 0.339·39-s − 0.153·41-s − 1.52·43-s − 0.961·45-s − 0.355·47-s − 0.208·49-s + 0.523·51-s + 0.293·53-s + 0.154·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 + 0.604T + 3T^{2} \) |
| 5 | \( 1 - 2.44T + 5T^{2} \) |
| 7 | \( 1 - 2.35T + 7T^{2} \) |
| 11 | \( 1 - 0.468T + 11T^{2} \) |
| 13 | \( 1 - 3.50T + 13T^{2} \) |
| 17 | \( 1 + 6.18T + 17T^{2} \) |
| 23 | \( 1 + 7.62T + 23T^{2} \) |
| 29 | \( 1 - 1.30T + 29T^{2} \) |
| 31 | \( 1 + 1.64T + 31T^{2} \) |
| 37 | \( 1 + 9.36T + 37T^{2} \) |
| 41 | \( 1 + 0.982T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 + 2.43T + 47T^{2} \) |
| 53 | \( 1 - 2.13T + 53T^{2} \) |
| 59 | \( 1 + 14.8T + 59T^{2} \) |
| 61 | \( 1 + 6.19T + 61T^{2} \) |
| 67 | \( 1 - 9.26T + 67T^{2} \) |
| 71 | \( 1 + 2.60T + 71T^{2} \) |
| 73 | \( 1 - 4.21T + 73T^{2} \) |
| 83 | \( 1 + 2.00T + 83T^{2} \) |
| 89 | \( 1 - 3.76T + 89T^{2} \) |
| 97 | \( 1 + 4.45T + 97T^{2} \) |
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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
−7.929727497572738355783550817825, −6.69422262954993921176163340925, −6.31120999980599225800133167917, −5.59010124536290363251790195586, −5.02102600306288352503478526535, −4.15869389139994752861573080138, −3.16751524979417497651927889043, −2.04366333495759929387551151368, −1.59195134373130681617193127889, 0,
1.59195134373130681617193127889, 2.04366333495759929387551151368, 3.16751524979417497651927889043, 4.15869389139994752861573080138, 5.02102600306288352503478526535, 5.59010124536290363251790195586, 6.31120999980599225800133167917, 6.69422262954993921176163340925, 7.929727497572738355783550817825
