| L(s) = 1 | + 5-s + 2·7-s − 3·9-s + 4·11-s + 4·13-s + 6·17-s + 19-s − 6·23-s + 25-s − 6·29-s + 8·31-s + 2·35-s + 4·37-s + 6·41-s − 2·43-s − 3·45-s + 10·47-s − 3·49-s + 4·55-s − 4·59-s + 2·61-s − 6·63-s + 4·65-s − 16·67-s + 8·71-s − 2·73-s + 8·77-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s − 9-s + 1.20·11-s + 1.10·13-s + 1.45·17-s + 0.229·19-s − 1.25·23-s + 1/5·25-s − 1.11·29-s + 1.43·31-s + 0.338·35-s + 0.657·37-s + 0.937·41-s − 0.304·43-s − 0.447·45-s + 1.45·47-s − 3/7·49-s + 0.539·55-s − 0.520·59-s + 0.256·61-s − 0.755·63-s + 0.496·65-s − 1.95·67-s + 0.949·71-s − 0.234·73-s + 0.911·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.793354984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.793354984\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.109697470666482338081825130066, −7.53656998764064509116704471904, −6.40848750007183407977569624761, −5.92750332898983200366820857073, −5.43301087582275963666841061427, −4.34019947108580430809121485032, −3.67893227757724672208259498621, −2.79616193352323122708053315928, −1.70464727676982945578966748826, −0.954837383298304925698123672196,
0.954837383298304925698123672196, 1.70464727676982945578966748826, 2.79616193352323122708053315928, 3.67893227757724672208259498621, 4.34019947108580430809121485032, 5.43301087582275963666841061427, 5.92750332898983200366820857073, 6.40848750007183407977569624761, 7.53656998764064509116704471904, 8.109697470666482338081825130066
