| L(s) = 1 | − 6·3-s − 34·7-s + 9·9-s − 16·11-s − 58·13-s + 70·17-s − 4·19-s + 204·21-s − 134·23-s + 108·27-s − 242·29-s − 100·31-s + 96·33-s + 438·37-s + 348·39-s − 138·41-s + 178·43-s + 22·47-s + 813·49-s − 420·51-s − 162·53-s + 24·57-s + 268·59-s + 250·61-s − 306·63-s + 422·67-s + 804·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.83·7-s + 1/3·9-s − 0.438·11-s − 1.23·13-s + 0.998·17-s − 0.0482·19-s + 2.11·21-s − 1.21·23-s + 0.769·27-s − 1.54·29-s − 0.579·31-s + 0.506·33-s + 1.94·37-s + 1.42·39-s − 0.525·41-s + 0.631·43-s + 0.0682·47-s + 2.37·49-s − 1.15·51-s − 0.419·53-s + 0.0557·57-s + 0.591·59-s + 0.524·61-s − 0.611·63-s + 0.769·67-s + 1.40·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3933823850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3933823850\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 7 | \( 1 + 34 T + p^{3} T^{2} \) |
| 11 | \( 1 + 16 T + p^{3} T^{2} \) |
| 13 | \( 1 + 58 T + p^{3} T^{2} \) |
| 17 | \( 1 - 70 T + p^{3} T^{2} \) |
| 19 | \( 1 + 4 T + p^{3} T^{2} \) |
| 23 | \( 1 + 134 T + p^{3} T^{2} \) |
| 29 | \( 1 + 242 T + p^{3} T^{2} \) |
| 31 | \( 1 + 100 T + p^{3} T^{2} \) |
| 37 | \( 1 - 438 T + p^{3} T^{2} \) |
| 41 | \( 1 + 138 T + p^{3} T^{2} \) |
| 43 | \( 1 - 178 T + p^{3} T^{2} \) |
| 47 | \( 1 - 22 T + p^{3} T^{2} \) |
| 53 | \( 1 + 162 T + p^{3} T^{2} \) |
| 59 | \( 1 - 268 T + p^{3} T^{2} \) |
| 61 | \( 1 - 250 T + p^{3} T^{2} \) |
| 67 | \( 1 - 422 T + p^{3} T^{2} \) |
| 71 | \( 1 - 12 p T + p^{3} T^{2} \) |
| 73 | \( 1 + 306 T + p^{3} T^{2} \) |
| 79 | \( 1 - 456 T + p^{3} T^{2} \) |
| 83 | \( 1 - 434 T + p^{3} T^{2} \) |
| 89 | \( 1 + 726 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1378 T + p^{3} T^{2} \) |
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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
−10.82939552059422632795407284134, −9.867623382948616724992733998851, −9.507078783228354434750101709140, −7.85044201709106938469081573509, −6.88868559934368095999891779123, −5.97174529941373289555949390127, −5.34484324472804247302437794463, −3.86027614037913168085825351878, −2.60458355937733393832297664879, −0.40507833065472674382357085346,
0.40507833065472674382357085346, 2.60458355937733393832297664879, 3.86027614037913168085825351878, 5.34484324472804247302437794463, 5.97174529941373289555949390127, 6.88868559934368095999891779123, 7.85044201709106938469081573509, 9.507078783228354434750101709140, 9.867623382948616724992733998851, 10.82939552059422632795407284134
