| L(s) = 1 | − 1.14·3-s + 1.71·5-s + 1.95·7-s − 1.69·9-s + 5.28·11-s + 4.62·13-s − 1.95·15-s − 1.34·19-s − 2.23·21-s − 6.44·23-s − 2.07·25-s + 5.36·27-s + 3.44·29-s + 2.64·31-s − 6.04·33-s + 3.34·35-s + 11.0·37-s − 5.28·39-s − 4.82·41-s + 5.75·43-s − 2.89·45-s + 6.84·47-s − 3.18·49-s − 8.09·53-s + 9.04·55-s + 1.54·57-s − 8.02·59-s + ⋯ |
| L(s) = 1 | − 0.660·3-s + 0.765·5-s + 0.738·7-s − 0.563·9-s + 1.59·11-s + 1.28·13-s − 0.505·15-s − 0.309·19-s − 0.487·21-s − 1.34·23-s − 0.414·25-s + 1.03·27-s + 0.639·29-s + 0.474·31-s − 1.05·33-s + 0.565·35-s + 1.81·37-s − 0.846·39-s − 0.753·41-s + 0.878·43-s − 0.431·45-s + 0.997·47-s − 0.455·49-s − 1.11·53-s + 1.21·55-s + 0.204·57-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.453927902\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.453927902\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 1.14T + 3T^{2} \) |
| 5 | \( 1 - 1.71T + 5T^{2} \) |
| 7 | \( 1 - 1.95T + 7T^{2} \) |
| 11 | \( 1 - 5.28T + 11T^{2} \) |
| 13 | \( 1 - 4.62T + 13T^{2} \) |
| 19 | \( 1 + 1.34T + 19T^{2} \) |
| 23 | \( 1 + 6.44T + 23T^{2} \) |
| 29 | \( 1 - 3.44T + 29T^{2} \) |
| 31 | \( 1 - 2.64T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 + 4.82T + 41T^{2} \) |
| 43 | \( 1 - 5.75T + 43T^{2} \) |
| 47 | \( 1 - 6.84T + 47T^{2} \) |
| 53 | \( 1 + 8.09T + 53T^{2} \) |
| 59 | \( 1 + 8.02T + 59T^{2} \) |
| 61 | \( 1 + 0.0929T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 - 0.564T + 73T^{2} \) |
| 79 | \( 1 - 6.26T + 79T^{2} \) |
| 83 | \( 1 - 9.12T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 1.29T + 97T^{2} \) |
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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
−7.958669692113030608265072224132, −6.63437883282712281690194716863, −6.22218770308307735948987533764, −5.93232198129889636914570655142, −5.02878519057161035155472705210, −4.24227268653903469341467463999, −3.62716963045857389191793441392, −2.44273571203242733444285970496, −1.59118515116492367622893337504, −0.843323705353058346644873634287,
0.843323705353058346644873634287, 1.59118515116492367622893337504, 2.44273571203242733444285970496, 3.62716963045857389191793441392, 4.24227268653903469341467463999, 5.02878519057161035155472705210, 5.93232198129889636914570655142, 6.22218770308307735948987533764, 6.63437883282712281690194716863, 7.958669692113030608265072224132
