| L(s) = 1 | + 1.38·2-s + 0.122·3-s − 0.0791·4-s + 0.133·5-s + 0.169·6-s − 2.88·8-s − 2.98·9-s + 0.184·10-s − 0.00969·12-s − 0.641·13-s + 0.0162·15-s − 3.83·16-s − 1.42·17-s − 4.13·18-s + 7.18·19-s − 0.0105·20-s + 1.66·23-s − 0.352·24-s − 4.98·25-s − 0.889·26-s − 0.733·27-s − 4.47·29-s + 0.0225·30-s + 6.83·31-s + 0.447·32-s − 1.97·34-s + 0.236·36-s + ⋯ |
| L(s) = 1 | + 0.980·2-s + 0.0707·3-s − 0.0395·4-s + 0.0594·5-s + 0.0693·6-s − 1.01·8-s − 0.994·9-s + 0.0582·10-s − 0.00279·12-s − 0.177·13-s + 0.00420·15-s − 0.958·16-s − 0.345·17-s − 0.975·18-s + 1.64·19-s − 0.00235·20-s + 0.347·23-s − 0.0720·24-s − 0.996·25-s − 0.174·26-s − 0.141·27-s − 0.831·29-s + 0.00412·30-s + 1.22·31-s + 0.0790·32-s − 0.338·34-s + 0.0393·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.287768439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.287768439\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.38T + 2T^{2} \) |
| 3 | \( 1 - 0.122T + 3T^{2} \) |
| 5 | \( 1 - 0.133T + 5T^{2} \) |
| 13 | \( 1 + 0.641T + 13T^{2} \) |
| 17 | \( 1 + 1.42T + 17T^{2} \) |
| 19 | \( 1 - 7.18T + 19T^{2} \) |
| 23 | \( 1 - 1.66T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 - 6.83T + 31T^{2} \) |
| 37 | \( 1 - 3.76T + 37T^{2} \) |
| 41 | \( 1 - 6.17T + 41T^{2} \) |
| 43 | \( 1 - 1.03T + 43T^{2} \) |
| 47 | \( 1 + 9.48T + 47T^{2} \) |
| 53 | \( 1 + 0.666T + 53T^{2} \) |
| 59 | \( 1 + 8.18T + 59T^{2} \) |
| 61 | \( 1 - 9.49T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 4.83T + 71T^{2} \) |
| 73 | \( 1 + 8.85T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 9.57T + 83T^{2} \) |
| 89 | \( 1 - 17.7T + 89T^{2} \) |
| 97 | \( 1 + 6.58T + 97T^{2} \) |
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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
−8.048994174148797877131953436057, −7.37163514889584068642731297538, −6.33946456027870780959499029698, −5.84268670765244316239645274956, −5.16915499950008629689872456338, −4.55892676836236376702889666985, −3.59075807020037059580645529190, −3.05483186883793930842182341866, −2.19956991539731351442218030035, −0.66461878615595190817919977385,
0.66461878615595190817919977385, 2.19956991539731351442218030035, 3.05483186883793930842182341866, 3.59075807020037059580645529190, 4.55892676836236376702889666985, 5.16915499950008629689872456338, 5.84268670765244316239645274956, 6.33946456027870780959499029698, 7.37163514889584068642731297538, 8.048994174148797877131953436057
