| L(s) = 1 | + 1.17·2-s − 0.620·4-s − 4.24·5-s − 3.49·7-s − 3.07·8-s − 4.98·10-s − 1.65·11-s − 4.59·13-s − 4.10·14-s − 2.37·16-s + 5.91·17-s + 1.31·19-s + 2.63·20-s − 1.94·22-s + 1.05·23-s + 13.0·25-s − 5.39·26-s + 2.16·28-s − 7.26·29-s − 2.73·31-s + 3.36·32-s + 6.94·34-s + 14.8·35-s − 3.39·37-s + 1.54·38-s + 13.0·40-s − 4.00·41-s + ⋯ |
| L(s) = 1 | + 0.830·2-s − 0.310·4-s − 1.89·5-s − 1.32·7-s − 1.08·8-s − 1.57·10-s − 0.499·11-s − 1.27·13-s − 1.09·14-s − 0.593·16-s + 1.43·17-s + 0.302·19-s + 0.588·20-s − 0.415·22-s + 0.219·23-s + 2.60·25-s − 1.05·26-s + 0.409·28-s − 1.34·29-s − 0.491·31-s + 0.594·32-s + 1.19·34-s + 2.50·35-s − 0.558·37-s + 0.250·38-s + 2.06·40-s − 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5086270193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5086270193\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + T \) |
| good | 2 | \( 1 - 1.17T + 2T^{2} \) |
| 5 | \( 1 + 4.24T + 5T^{2} \) |
| 7 | \( 1 + 3.49T + 7T^{2} \) |
| 11 | \( 1 + 1.65T + 11T^{2} \) |
| 13 | \( 1 + 4.59T + 13T^{2} \) |
| 17 | \( 1 - 5.91T + 17T^{2} \) |
| 19 | \( 1 - 1.31T + 19T^{2} \) |
| 23 | \( 1 - 1.05T + 23T^{2} \) |
| 29 | \( 1 + 7.26T + 29T^{2} \) |
| 31 | \( 1 + 2.73T + 31T^{2} \) |
| 37 | \( 1 + 3.39T + 37T^{2} \) |
| 41 | \( 1 + 4.00T + 41T^{2} \) |
| 43 | \( 1 + 0.430T + 43T^{2} \) |
| 47 | \( 1 - 2.62T + 47T^{2} \) |
| 53 | \( 1 - 2.43T + 53T^{2} \) |
| 59 | \( 1 - 8.73T + 59T^{2} \) |
| 61 | \( 1 + 5.43T + 61T^{2} \) |
| 67 | \( 1 + 3.16T + 67T^{2} \) |
| 71 | \( 1 - 7.09T + 71T^{2} \) |
| 73 | \( 1 - 1.45T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 - 1.76T + 83T^{2} \) |
| 89 | \( 1 + 7.16T + 89T^{2} \) |
| 97 | \( 1 - 18.8T + 97T^{2} \) |
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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
−9.105241604328166503062454742111, −8.149121463428659648762835050215, −7.43317355426947737991039468605, −6.88123684156618822209332837128, −5.62981645294764706851222515207, −5.01839519903748286409069604748, −4.02563642586635774067818639977, −3.43365990838329735472914627859, −2.87202945134378618182893045843, −0.38892071632581215694884536117,
0.38892071632581215694884536117, 2.87202945134378618182893045843, 3.43365990838329735472914627859, 4.02563642586635774067818639977, 5.01839519903748286409069604748, 5.62981645294764706851222515207, 6.88123684156618822209332837128, 7.43317355426947737991039468605, 8.149121463428659648762835050215, 9.105241604328166503062454742111
