| L(s) = 1 | + 0.381·2-s − 0.381·3-s − 1.85·4-s + 0.381·5-s − 0.145·6-s + 7-s − 1.47·8-s − 2.85·9-s + 0.145·10-s − 4.85·11-s + 0.708·12-s + 0.381·14-s − 0.145·15-s + 3.14·16-s + 7.47·17-s − 1.09·18-s + 4.85·19-s − 0.708·20-s − 0.381·21-s − 1.85·22-s + 4.47·23-s + 0.562·24-s − 4.85·25-s + 2.23·27-s − 1.85·28-s − 4.09·29-s − 0.0557·30-s + ⋯ |
| L(s) = 1 | + 0.270·2-s − 0.220·3-s − 0.927·4-s + 0.170·5-s − 0.0595·6-s + 0.377·7-s − 0.520·8-s − 0.951·9-s + 0.0461·10-s − 1.46·11-s + 0.204·12-s + 0.102·14-s − 0.0376·15-s + 0.786·16-s + 1.81·17-s − 0.256·18-s + 1.11·19-s − 0.158·20-s − 0.0833·21-s − 0.395·22-s + 0.932·23-s + 0.114·24-s − 0.970·25-s + 0.430·27-s − 0.350·28-s − 0.759·29-s − 0.0101·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.228550007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.228550007\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.381T + 2T^{2} \) |
| 3 | \( 1 + 0.381T + 3T^{2} \) |
| 5 | \( 1 - 0.381T + 5T^{2} \) |
| 11 | \( 1 + 4.85T + 11T^{2} \) |
| 17 | \( 1 - 7.47T + 17T^{2} \) |
| 19 | \( 1 - 4.85T + 19T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 29 | \( 1 + 4.09T + 29T^{2} \) |
| 31 | \( 1 - 8.70T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 5.23T + 41T^{2} \) |
| 43 | \( 1 + 7.56T + 43T^{2} \) |
| 47 | \( 1 - 2.23T + 47T^{2} \) |
| 53 | \( 1 - 8.23T + 53T^{2} \) |
| 59 | \( 1 - 2.23T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 0.708T + 67T^{2} \) |
| 71 | \( 1 - 8.18T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 6.70T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−9.842611353272889075000282340984, −8.960171870875045147957662254235, −7.984410843302600345325203557243, −7.65517688832451792158966327856, −6.00701635809046028972676613715, −5.38394149739157780591913072623, −4.92170792386454532261323230260, −3.52684003627174272452693858405, −2.71551082361191645407167010437, −0.814714983260082074970639749580,
0.814714983260082074970639749580, 2.71551082361191645407167010437, 3.52684003627174272452693858405, 4.92170792386454532261323230260, 5.38394149739157780591913072623, 6.00701635809046028972676613715, 7.65517688832451792158966327856, 7.984410843302600345325203557243, 8.960171870875045147957662254235, 9.842611353272889075000282340984
