| L(s) = 1 | − 2·3-s + 9-s + 11-s + 2·13-s − 7·17-s − 3·23-s + 4·27-s + 10·29-s − 9·31-s − 2·33-s − 4·37-s − 4·39-s + 9·41-s + 2·43-s + 7·47-s + 14·51-s + 6·53-s + 12·59-s − 4·61-s + 14·67-s + 6·69-s − 3·71-s + 6·73-s + 11·79-s − 11·81-s + 6·83-s − 20·87-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 1.69·17-s − 0.625·23-s + 0.769·27-s + 1.85·29-s − 1.61·31-s − 0.348·33-s − 0.657·37-s − 0.640·39-s + 1.40·41-s + 0.304·43-s + 1.02·47-s + 1.96·51-s + 0.824·53-s + 1.56·59-s − 0.512·61-s + 1.71·67-s + 0.722·69-s − 0.356·71-s + 0.702·73-s + 1.23·79-s − 1.22·81-s + 0.658·83-s − 2.14·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 215600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.880564148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.880564148\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−12.85208303923052, −12.43520684303743, −12.04751130519224, −11.53574602301295, −11.06977762380120, −10.84103798695116, −10.35563231913568, −9.853232905147621, −9.037495346809339, −8.888564336452389, −8.366569903272878, −7.687973706059488, −7.095045378545642, −6.616757115234871, −6.276782015449655, −5.832590700275230, −5.209708131030229, −4.844296494041123, −4.101607532196602, −3.886939228282981, −3.016965111116448, −2.263027109762102, −1.888003664151607, −0.7209088210712259, −0.6328497596570467,
0.6328497596570467, 0.7209088210712259, 1.888003664151607, 2.263027109762102, 3.016965111116448, 3.886939228282981, 4.101607532196602, 4.844296494041123, 5.209708131030229, 5.832590700275230, 6.276782015449655, 6.616757115234871, 7.095045378545642, 7.687973706059488, 8.366569903272878, 8.888564336452389, 9.037495346809339, 9.853232905147621, 10.35563231913568, 10.84103798695116, 11.06977762380120, 11.53574602301295, 12.04751130519224, 12.43520684303743, 12.85208303923052
