L(s) = 1 | − 0.983·2-s − 1.57·3-s − 1.03·4-s − 0.398·5-s + 1.54·6-s − 7-s + 2.98·8-s − 0.529·9-s + 0.391·10-s − 4.24·11-s + 1.62·12-s + 0.983·14-s + 0.626·15-s − 0.870·16-s − 5.10·17-s + 0.521·18-s − 2.12·19-s + 0.411·20-s + 1.57·21-s + 4.17·22-s + 2.19·23-s − 4.68·24-s − 4.84·25-s + 5.54·27-s + 1.03·28-s + 2.90·29-s − 0.616·30-s + ⋯ |
L(s) = 1 | − 0.695·2-s − 0.907·3-s − 0.516·4-s − 0.178·5-s + 0.631·6-s − 0.377·7-s + 1.05·8-s − 0.176·9-s + 0.123·10-s − 1.27·11-s + 0.468·12-s + 0.262·14-s + 0.161·15-s − 0.217·16-s − 1.23·17-s + 0.122·18-s − 0.487·19-s + 0.0919·20-s + 0.342·21-s + 0.889·22-s + 0.457·23-s − 0.957·24-s − 0.968·25-s + 1.06·27-s + 0.195·28-s + 0.540·29-s − 0.112·30-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
\(L(1)\) |
\(\approx\) |
\(0.2350268384\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2350268384\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.983T + 2T^{2} \) |
| 3 | \( 1 + 1.57T + 3T^{2} \) |
| 5 | \( 1 + 0.398T + 5T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 17 | \( 1 + 5.10T + 17T^{2} \) |
| 19 | \( 1 + 2.12T + 19T^{2} \) |
| 23 | \( 1 - 2.19T + 23T^{2} \) |
| 29 | \( 1 - 2.90T + 29T^{2} \) |
| 31 | \( 1 + 2.20T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 + 5.07T + 41T^{2} \) |
| 43 | \( 1 + 0.328T + 43T^{2} \) |
| 47 | \( 1 + 6.62T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 - 7.70T + 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 + 1.22T + 67T^{2} \) |
| 71 | \( 1 - 1.75T + 71T^{2} \) |
| 73 | \( 1 - 6.11T + 73T^{2} \) |
| 79 | \( 1 - 4.20T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−9.961671609076192442055579666077, −8.758731440218873033758542380536, −8.416534423232833999099136623353, −7.31431344276228102161713069820, −6.52058980976444262278731765828, −5.38627219345914994342854915514, −4.86952433825284308669194594905, −3.68298679720805545410519411078, −2.21075076459847179832526550392, −0.40164968904711270123471872210,
0.40164968904711270123471872210, 2.21075076459847179832526550392, 3.68298679720805545410519411078, 4.86952433825284308669194594905, 5.38627219345914994342854915514, 6.52058980976444262278731765828, 7.31431344276228102161713069820, 8.416534423232833999099136623353, 8.758731440218873033758542380536, 9.961671609076192442055579666077