| L(s) = 1 | − 1.36·2-s − 3-s − 0.138·4-s − 3.32·5-s + 1.36·6-s − 7-s + 2.91·8-s + 9-s + 4.53·10-s + 1.11·11-s + 0.138·12-s + 1.36·14-s + 3.32·15-s − 3.70·16-s + 0.665·17-s − 1.36·18-s + 3.22·19-s + 0.458·20-s + 21-s − 1.51·22-s + 5.07·23-s − 2.91·24-s + 6.02·25-s − 27-s + 0.138·28-s − 0.615·29-s − 4.53·30-s + ⋯ |
| L(s) = 1 | − 0.964·2-s − 0.577·3-s − 0.0690·4-s − 1.48·5-s + 0.557·6-s − 0.377·7-s + 1.03·8-s + 0.333·9-s + 1.43·10-s + 0.335·11-s + 0.0398·12-s + 0.364·14-s + 0.857·15-s − 0.926·16-s + 0.161·17-s − 0.321·18-s + 0.740·19-s + 0.102·20-s + 0.218·21-s − 0.324·22-s + 1.05·23-s − 0.595·24-s + 1.20·25-s − 0.192·27-s + 0.0261·28-s − 0.114·29-s − 0.827·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3518583434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3518583434\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.36T + 2T^{2} \) |
| 5 | \( 1 + 3.32T + 5T^{2} \) |
| 11 | \( 1 - 1.11T + 11T^{2} \) |
| 17 | \( 1 - 0.665T + 17T^{2} \) |
| 19 | \( 1 - 3.22T + 19T^{2} \) |
| 23 | \( 1 - 5.07T + 23T^{2} \) |
| 29 | \( 1 + 0.615T + 29T^{2} \) |
| 31 | \( 1 + 8.84T + 31T^{2} \) |
| 37 | \( 1 + 3.79T + 37T^{2} \) |
| 41 | \( 1 - 0.916T + 41T^{2} \) |
| 43 | \( 1 - 8.09T + 43T^{2} \) |
| 47 | \( 1 + 13.0T + 47T^{2} \) |
| 53 | \( 1 - 3.52T + 53T^{2} \) |
| 59 | \( 1 - 5.48T + 59T^{2} \) |
| 61 | \( 1 + 14.1T + 61T^{2} \) |
| 67 | \( 1 + 3.47T + 67T^{2} \) |
| 71 | \( 1 - 4.77T + 71T^{2} \) |
| 73 | \( 1 + 7.86T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 + 1.42T + 97T^{2} \) |
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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
−8.736216496775405276139616256421, −7.58422074798024713172416194573, −7.49721589321144736844836087903, −6.64801736183355672276067483389, −5.49460725932256146242477040811, −4.69432529264963244911527930731, −3.93231717312042436385909792160, −3.17294307055690495076610215020, −1.48337139677062981704276590571, −0.44619932091872179528723609212,
0.44619932091872179528723609212, 1.48337139677062981704276590571, 3.17294307055690495076610215020, 3.93231717312042436385909792160, 4.69432529264963244911527930731, 5.49460725932256146242477040811, 6.64801736183355672276067483389, 7.49721589321144736844836087903, 7.58422074798024713172416194573, 8.736216496775405276139616256421
