| L(s) = 1 | + 16·5-s − 28·7-s + 34·11-s − 13·13-s − 138·17-s − 108·19-s − 52·23-s + 131·25-s + 190·29-s + 176·31-s − 448·35-s + 342·37-s − 240·41-s + 140·43-s + 454·47-s + 441·49-s − 198·53-s + 544·55-s − 154·59-s + 34·61-s − 208·65-s + 656·67-s + 550·71-s + 614·73-s − 952·77-s − 8·79-s + 762·83-s + ⋯ |
| L(s) = 1 | + 1.43·5-s − 1.51·7-s + 0.931·11-s − 0.277·13-s − 1.96·17-s − 1.30·19-s − 0.471·23-s + 1.04·25-s + 1.21·29-s + 1.01·31-s − 2.16·35-s + 1.51·37-s − 0.914·41-s + 0.496·43-s + 1.40·47-s + 9/7·49-s − 0.513·53-s + 1.33·55-s − 0.339·59-s + 0.0713·61-s − 0.396·65-s + 1.19·67-s + 0.919·71-s + 0.984·73-s − 1.40·77-s − 0.0113·79-s + 1.00·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.090123732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.090123732\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + p T \) |
| good | 5 | \( 1 - 16 T + p^{3} T^{2} \) |
| 7 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 34 T + p^{3} T^{2} \) |
| 17 | \( 1 + 138 T + p^{3} T^{2} \) |
| 19 | \( 1 + 108 T + p^{3} T^{2} \) |
| 23 | \( 1 + 52 T + p^{3} T^{2} \) |
| 29 | \( 1 - 190 T + p^{3} T^{2} \) |
| 31 | \( 1 - 176 T + p^{3} T^{2} \) |
| 37 | \( 1 - 342 T + p^{3} T^{2} \) |
| 41 | \( 1 + 240 T + p^{3} T^{2} \) |
| 43 | \( 1 - 140 T + p^{3} T^{2} \) |
| 47 | \( 1 - 454 T + p^{3} T^{2} \) |
| 53 | \( 1 + 198 T + p^{3} T^{2} \) |
| 59 | \( 1 + 154 T + p^{3} T^{2} \) |
| 61 | \( 1 - 34 T + p^{3} T^{2} \) |
| 67 | \( 1 - 656 T + p^{3} T^{2} \) |
| 71 | \( 1 - 550 T + p^{3} T^{2} \) |
| 73 | \( 1 - 614 T + p^{3} T^{2} \) |
| 79 | \( 1 + 8 T + p^{3} T^{2} \) |
| 83 | \( 1 - 762 T + p^{3} T^{2} \) |
| 89 | \( 1 - 444 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1022 T + p^{3} T^{2} \) |
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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
−9.153783204566902111820325221224, −8.329023603381918006652985013027, −6.82006329789906365421283936615, −6.39920708926293870666574683470, −6.09658346351003424337354943804, −4.74767910479822582247339848330, −3.93779674113940551225010265874, −2.64208343378806915709390516261, −2.10152691088045426321000263779, −0.64969236614616481216090046033,
0.64969236614616481216090046033, 2.10152691088045426321000263779, 2.64208343378806915709390516261, 3.93779674113940551225010265874, 4.74767910479822582247339848330, 6.09658346351003424337354943804, 6.39920708926293870666574683470, 6.82006329789906365421283936615, 8.329023603381918006652985013027, 9.153783204566902111820325221224
