| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 2·7-s − 8-s + 9-s − 12-s + 4·13-s + 2·14-s + 16-s − 6·17-s − 18-s + 8·19-s + 2·21-s + 24-s − 4·26-s − 27-s − 2·28-s + 31-s − 32-s + 6·34-s + 36-s + 4·37-s − 8·38-s − 4·39-s − 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s + 1.10·13-s + 0.534·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 1.83·19-s + 0.436·21-s + 0.204·24-s − 0.784·26-s − 0.192·27-s − 0.377·28-s + 0.179·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s + 0.657·37-s − 1.29·38-s − 0.640·39-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9074482390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9074482390\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.460068752185368968389030422776, −7.47689438163177606510948862438, −6.89603122452971426718015346961, −6.22288671143227845702974054753, −5.62897566964802008811404209196, −4.63899227950574788419618160684, −3.64766166759127670855842491787, −2.86396218640878501834488499395, −1.64997389686777790527081002256, −0.61974766652524967827378665150,
0.61974766652524967827378665150, 1.64997389686777790527081002256, 2.86396218640878501834488499395, 3.64766166759127670855842491787, 4.63899227950574788419618160684, 5.62897566964802008811404209196, 6.22288671143227845702974054753, 6.89603122452971426718015346961, 7.47689438163177606510948862438, 8.460068752185368968389030422776
