| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 6·6-s + 7·7-s − 8·8-s + 9·9-s − 69·11-s − 12·12-s − 64·13-s − 14·14-s + 16·16-s − 114·17-s − 18·18-s + 56·19-s − 21·21-s + 138·22-s + 9·23-s + 24·24-s + 128·26-s − 27·27-s + 28·28-s − 33·29-s − 70·31-s − 32·32-s + 207·33-s + 228·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.89·11-s − 0.288·12-s − 1.36·13-s − 0.267·14-s + 1/4·16-s − 1.62·17-s − 0.235·18-s + 0.676·19-s − 0.218·21-s + 1.33·22-s + 0.0815·23-s + 0.204·24-s + 0.965·26-s − 0.192·27-s + 0.188·28-s − 0.211·29-s − 0.405·31-s − 0.176·32-s + 1.09·33-s + 1.15·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4670794115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4670794115\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 11 | \( 1 + 69 T + p^{3} T^{2} \) |
| 13 | \( 1 + 64 T + p^{3} T^{2} \) |
| 17 | \( 1 + 114 T + p^{3} T^{2} \) |
| 19 | \( 1 - 56 T + p^{3} T^{2} \) |
| 23 | \( 1 - 9 T + p^{3} T^{2} \) |
| 29 | \( 1 + 33 T + p^{3} T^{2} \) |
| 31 | \( 1 + 70 T + p^{3} T^{2} \) |
| 37 | \( 1 - 53 T + p^{3} T^{2} \) |
| 41 | \( 1 - 504 T + p^{3} T^{2} \) |
| 43 | \( 1 - 137 T + p^{3} T^{2} \) |
| 47 | \( 1 + 600 T + p^{3} T^{2} \) |
| 53 | \( 1 + 570 T + p^{3} T^{2} \) |
| 59 | \( 1 - 48 T + p^{3} T^{2} \) |
| 61 | \( 1 - 524 T + p^{3} T^{2} \) |
| 67 | \( 1 + T + p^{3} T^{2} \) |
| 71 | \( 1 + 93 T + p^{3} T^{2} \) |
| 73 | \( 1 + 850 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1261 T + p^{3} T^{2} \) |
| 83 | \( 1 - 486 T + p^{3} T^{2} \) |
| 89 | \( 1 - 300 T + p^{3} T^{2} \) |
| 97 | \( 1 - 866 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.672573431880193469304417554060, −8.727732292603924876154592289295, −7.66333067544613336330393581324, −7.36255347826944308566057486657, −6.18813734658902622946243528755, −5.19556697591510938470754368806, −4.57367333064461801575978725969, −2.83271687120510216524247827615, −2.01683444397926651254278670100, −0.38895468006956234136415157668,
0.38895468006956234136415157668, 2.01683444397926651254278670100, 2.83271687120510216524247827615, 4.57367333064461801575978725969, 5.19556697591510938470754368806, 6.18813734658902622946243528755, 7.36255347826944308566057486657, 7.66333067544613336330393581324, 8.727732292603924876154592289295, 9.672573431880193469304417554060
