| L(s) = 1 | − 2.04·2-s − 3-s + 2.18·4-s − 0.155·5-s + 2.04·6-s − 0.385·8-s + 9-s + 0.317·10-s − 4.80·11-s − 2.18·12-s − 13-s + 0.155·15-s − 3.58·16-s + 7.14·17-s − 2.04·18-s − 4.58·19-s − 0.339·20-s + 9.84·22-s + 0.871·23-s + 0.385·24-s − 4.97·25-s + 2.04·26-s − 27-s + 9.59·29-s − 0.317·30-s − 6.65·31-s + 8.11·32-s + ⋯ |
| L(s) = 1 | − 1.44·2-s − 0.577·3-s + 1.09·4-s − 0.0693·5-s + 0.835·6-s − 0.136·8-s + 0.333·9-s + 0.100·10-s − 1.45·11-s − 0.631·12-s − 0.277·13-s + 0.0400·15-s − 0.896·16-s + 1.73·17-s − 0.482·18-s − 1.05·19-s − 0.0758·20-s + 2.09·22-s + 0.181·23-s + 0.0787·24-s − 0.995·25-s + 0.401·26-s − 0.192·27-s + 1.78·29-s − 0.0579·30-s − 1.19·31-s + 1.43·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4362771624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4362771624\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 2.04T + 2T^{2} \) |
| 5 | \( 1 + 0.155T + 5T^{2} \) |
| 11 | \( 1 + 4.80T + 11T^{2} \) |
| 17 | \( 1 - 7.14T + 17T^{2} \) |
| 19 | \( 1 + 4.58T + 19T^{2} \) |
| 23 | \( 1 - 0.871T + 23T^{2} \) |
| 29 | \( 1 - 9.59T + 29T^{2} \) |
| 31 | \( 1 + 6.65T + 31T^{2} \) |
| 37 | \( 1 - 1.05T + 37T^{2} \) |
| 41 | \( 1 - 5.18T + 41T^{2} \) |
| 43 | \( 1 - 4.27T + 43T^{2} \) |
| 47 | \( 1 + 8.69T + 47T^{2} \) |
| 53 | \( 1 + 1.90T + 53T^{2} \) |
| 59 | \( 1 + 8.21T + 59T^{2} \) |
| 61 | \( 1 + 6.18T + 61T^{2} \) |
| 67 | \( 1 - 6.06T + 67T^{2} \) |
| 71 | \( 1 + 0.313T + 71T^{2} \) |
| 73 | \( 1 - 0.964T + 73T^{2} \) |
| 79 | \( 1 - 5.96T + 79T^{2} \) |
| 83 | \( 1 + 5.37T + 83T^{2} \) |
| 89 | \( 1 - 16.3T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−9.380555137553664953057877889195, −8.211341633424605854773009858165, −7.905920026651693054270347355115, −7.16622103973968422020203467811, −6.16672118181677372944872916851, −5.31217109596806587656075531398, −4.41772053390734449629283548798, −2.98370999149578798563000999628, −1.84856617627276058254069744879, −0.56196759167419549044524044573,
0.56196759167419549044524044573, 1.84856617627276058254069744879, 2.98370999149578798563000999628, 4.41772053390734449629283548798, 5.31217109596806587656075531398, 6.16672118181677372944872916851, 7.16622103973968422020203467811, 7.905920026651693054270347355115, 8.211341633424605854773009858165, 9.380555137553664953057877889195
