L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 2·7-s − 8-s + 9-s + 6·11-s − 12-s + 4·13-s + 2·14-s + 16-s + 6·17-s − 18-s + 19-s + 2·21-s − 6·22-s + 24-s − 4·26-s − 27-s − 2·28-s + 8·31-s − 32-s − 6·33-s − 6·34-s + 36-s − 8·37-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 + 1.80·11-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 + 0.229·19-s + 0.436·21-s − 1.27·22-s + 0.204·24-s − 0.784·26-s − 0.192·27-s − 0.377·28-s + 1.43·31-s − 0.176·32-s − 1.04·33-s − 1.02·34-s + 1/6·36-s − 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.226963296\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.226963296\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−8.777277048365311866300196876149, −8.210451136132986552014643220057, −7.10001927344968593094115378805, −6.51279669961772372427076950654, −6.05564997334095721817435593795, −5.04652259411543408267351447418, −3.74402078559359188479709389503, −3.30877212461539318698457634531, −1.63352354205036066782550504517, −0.852684977198359607547549426954,
0.852684977198359607547549426954, 1.63352354205036066782550504517, 3.30877212461539318698457634531, 3.74402078559359188479709389503, 5.04652259411543408267351447418, 6.05564997334095721817435593795, 6.51279669961772372427076950654, 7.10001927344968593094115378805, 8.210451136132986552014643220057, 8.777277048365311866300196876149