L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 4·7-s + 8-s + 9-s + 6·11-s − 12-s − 2·13-s + 4·14-s + 16-s + 6·17-s + 18-s + 2·19-s − 4·21-s + 6·22-s − 24-s − 2·26-s − 27-s + 4·28-s + 6·29-s + 8·31-s + 32-s − 6·33-s + 6·34-s + 36-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 1.80·11-s − 0.288·12-s − 0.554·13-s + 1.06·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.458·19-s − 0.872·21-s + 1.27·22-s − 0.204·24-s − 0.392·26-s − 0.192·27-s + 0.755·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s − 1.04·33-s + 1.02·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.970487310\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.970487310\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 37 | \( 1 + T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−8.055726123098444627886230125545, −7.30705620941724579488862516901, −6.58554524891292074595415555509, −5.97145120745249917270090907669, −4.96630665114720263978232935941, −4.77771273434562900846990872627, −3.85467227058861641537871591999, −2.98384225780378724401143323283, −1.61482948377529459692615710851, −1.17258869157908877745007323261,
1.17258869157908877745007323261, 1.61482948377529459692615710851, 2.98384225780378724401143323283, 3.85467227058861641537871591999, 4.77771273434562900846990872627, 4.96630665114720263978232935941, 5.97145120745249917270090907669, 6.58554524891292074595415555509, 7.30705620941724579488862516901, 8.055726123098444627886230125545