| L(s) = 1 | − 2.68·2-s − 2.90·3-s + 5.22·4-s − 3.09·5-s + 7.82·6-s − 1.70·7-s − 8.66·8-s + 5.46·9-s + 8.31·10-s − 11-s − 15.2·12-s + 4.58·14-s + 9.00·15-s + 12.8·16-s + 1.04·17-s − 14.6·18-s + 0.291·19-s − 16.1·20-s + 4.95·21-s + 2.68·22-s − 7.88·23-s + 25.2·24-s + 4.57·25-s − 7.17·27-s − 8.90·28-s − 0.935·29-s − 24.1·30-s + ⋯ |
| L(s) = 1 | − 1.90·2-s − 1.68·3-s + 2.61·4-s − 1.38·5-s + 3.19·6-s − 0.644·7-s − 3.06·8-s + 1.82·9-s + 2.62·10-s − 0.301·11-s − 4.38·12-s + 1.22·14-s + 2.32·15-s + 3.21·16-s + 0.252·17-s − 3.46·18-s + 0.0667·19-s − 3.61·20-s + 1.08·21-s + 0.573·22-s − 1.64·23-s + 5.14·24-s + 0.914·25-s − 1.38·27-s − 1.68·28-s − 0.173·29-s − 4.41·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02367841638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02367841638\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.68T + 2T^{2} \) |
| 3 | \( 1 + 2.90T + 3T^{2} \) |
| 5 | \( 1 + 3.09T + 5T^{2} \) |
| 7 | \( 1 + 1.70T + 7T^{2} \) |
| 17 | \( 1 - 1.04T + 17T^{2} \) |
| 19 | \( 1 - 0.291T + 19T^{2} \) |
| 23 | \( 1 + 7.88T + 23T^{2} \) |
| 29 | \( 1 + 0.935T + 29T^{2} \) |
| 31 | \( 1 + 1.01T + 31T^{2} \) |
| 37 | \( 1 - 6.79T + 37T^{2} \) |
| 41 | \( 1 + 4.76T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + 2.04T + 47T^{2} \) |
| 53 | \( 1 + 4.76T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 - 9.48T + 61T^{2} \) |
| 67 | \( 1 + 9.43T + 67T^{2} \) |
| 71 | \( 1 + 0.582T + 71T^{2} \) |
| 73 | \( 1 + 15.8T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 + 5.79T + 89T^{2} \) |
| 97 | \( 1 - 0.922T + 97T^{2} \) |
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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
−9.439991346179112997070852487821, −8.248097166610262014648100595005, −7.80169615029338838433302299093, −6.98333053505210273082302943403, −6.39067264923886925875108244983, −5.61385553609829250739892167073, −4.30878144937850969541029703681, −3.12557056049352970992569476041, −1.50508800305764262721069127965, −0.15348015517027209637405365755,
0.15348015517027209637405365755, 1.50508800305764262721069127965, 3.12557056049352970992569476041, 4.30878144937850969541029703681, 5.61385553609829250739892167073, 6.39067264923886925875108244983, 6.98333053505210273082302943403, 7.80169615029338838433302299093, 8.248097166610262014648100595005, 9.439991346179112997070852487821
