| L(s) = 1 | + 3·5-s − 5·7-s − 3·9-s − 5·11-s − 4·13-s − 3·17-s + 19-s + 4·25-s + 10·31-s − 15·35-s + 8·37-s − 5·43-s − 9·45-s + 5·47-s + 18·49-s − 6·53-s − 15·55-s − 10·59-s − 5·61-s + 15·63-s − 12·65-s − 10·67-s + 10·71-s − 11·73-s + 25·77-s − 10·79-s + 9·81-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.88·7-s − 9-s − 1.50·11-s − 1.10·13-s − 0.727·17-s + 0.229·19-s + 4/5·25-s + 1.79·31-s − 2.53·35-s + 1.31·37-s − 0.762·43-s − 1.34·45-s + 0.729·47-s + 18/7·49-s − 0.824·53-s − 2.02·55-s − 1.30·59-s − 0.640·61-s + 1.88·63-s − 1.48·65-s − 1.22·67-s + 1.18·71-s − 1.28·73-s + 2.84·77-s − 1.12·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−9.932004732623641143909868708294, −9.660921116637740874206419270015, −8.657607798255729945382040282859, −7.44006890990342389833161815303, −6.31101389249620377592400624257, −5.86241105176971199112628899877, −4.80406574906688719619549586116, −2.88174055098079092197095933553, −2.57940326354661234114682009216, 0,
2.57940326354661234114682009216, 2.88174055098079092197095933553, 4.80406574906688719619549586116, 5.86241105176971199112628899877, 6.31101389249620377592400624257, 7.44006890990342389833161815303, 8.657607798255729945382040282859, 9.660921116637740874206419270015, 9.932004732623641143909868708294
