| L(s) = 1 | − 2·5-s − 11-s − 2·13-s + 4·17-s + 6·19-s − 25-s − 8·29-s − 8·31-s − 10·37-s + 8·41-s − 2·43-s + 8·47-s − 2·53-s + 2·55-s + 12·59-s + 10·61-s + 4·65-s + 12·67-s + 8·71-s − 6·73-s + 2·79-s + 16·83-s − 8·85-s − 14·89-s − 12·95-s + 2·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.301·11-s − 0.554·13-s + 0.970·17-s + 1.37·19-s − 1/5·25-s − 1.48·29-s − 1.43·31-s − 1.64·37-s + 1.24·41-s − 0.304·43-s + 1.16·47-s − 0.274·53-s + 0.269·55-s + 1.56·59-s + 1.28·61-s + 0.496·65-s + 1.46·67-s + 0.949·71-s − 0.702·73-s + 0.225·79-s + 1.75·83-s − 0.867·85-s − 1.48·89-s − 1.23·95-s + 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.876476635\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.876476635\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−12.51615747320801, −12.27406055866457, −11.64345269716502, −11.43255079183603, −10.90044180467436, −10.41596030616477, −9.815584459257598, −9.512221706514681, −9.046220749329688, −8.392490721289887, −7.963173136331838, −7.477593754524590, −7.212477174482656, −6.858558821073207, −5.836907257816594, −5.497856989208291, −5.246949544554299, −4.540470371859872, −3.783563960835769, −3.601847715242882, −3.136810020806651, −2.234832782681044, −1.889262342434736, −0.9083963222133056, −0.4432112108933114,
0.4432112108933114, 0.9083963222133056, 1.889262342434736, 2.234832782681044, 3.136810020806651, 3.601847715242882, 3.783563960835769, 4.540470371859872, 5.246949544554299, 5.497856989208291, 5.836907257816594, 6.858558821073207, 7.212477174482656, 7.477593754524590, 7.963173136331838, 8.392490721289887, 9.046220749329688, 9.512221706514681, 9.815584459257598, 10.41596030616477, 10.90044180467436, 11.43255079183603, 11.64345269716502, 12.27406055866457, 12.51615747320801
