| L(s) = 1 | − 2-s + 3-s − 4-s − 6-s + 3·8-s + 9-s − 12-s + 13-s − 16-s + 2·17-s − 18-s − 19-s − 2·23-s + 3·24-s − 26-s + 27-s + 9·29-s − 9·31-s − 5·32-s − 2·34-s − 36-s + 10·37-s + 38-s + 39-s + 8·43-s + 2·46-s + 7·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.288·12-s + 0.277·13-s − 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.229·19-s − 0.417·23-s + 0.612·24-s − 0.196·26-s + 0.192·27-s + 1.67·29-s − 1.61·31-s − 0.883·32-s − 0.342·34-s − 1/6·36-s + 1.64·37-s + 0.162·38-s + 0.160·39-s + 1.21·43-s + 0.294·46-s + 1.02·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.914985549\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.914985549\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.53102147190883, −14.06710995341055, −13.64781830177288, −13.03904138034414, −12.63722427562620, −12.08557776203915, −11.33930268461403, −10.71263309577151, −10.38419849084164, −9.717940194270058, −9.306141136736812, −8.886208190504084, −8.195287091393801, −7.954677618215023, −7.330738357505322, −6.750338328201620, −5.959742275165233, −5.395549725201448, −4.636316274609156, −4.094364535969443, −3.605771119383392, −2.714143410440049, −2.088047957681049, −1.195378768827389, −0.6113487926805603,
0.6113487926805603, 1.195378768827389, 2.088047957681049, 2.714143410440049, 3.605771119383392, 4.094364535969443, 4.636316274609156, 5.395549725201448, 5.959742275165233, 6.750338328201620, 7.330738357505322, 7.954677618215023, 8.195287091393801, 8.886208190504084, 9.306141136736812, 9.717940194270058, 10.38419849084164, 10.71263309577151, 11.33930268461403, 12.08557776203915, 12.63722427562620, 13.03904138034414, 13.64781830177288, 14.06710995341055, 14.53102147190883
