L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 2·11-s + 12-s + 13-s + 14-s − 15-s + 16-s − 3·17-s + 18-s + 2·19-s − 20-s + 21-s + 2·22-s + 3·23-s + 24-s + 25-s + 26-s + 27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 0.458·19-s − 0.223·20-s + 0.218·21-s + 0.426·22-s + 0.625·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.086039370\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.086039370\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−12.63397935572917, −12.18490542426713, −11.78551827499263, −11.24342331722858, −10.92851597745078, −10.51918639932818, −9.846505951754306, −9.210382794628510, −9.052331108005281, −8.431835600256164, −7.871990763085844, −7.592179509858635, −6.977595079376814, −6.558576087818673, −6.149809767351052, −5.342724390704455, −5.058943200332452, −4.395706144017710, −3.954915430144945, −3.627627063877335, −2.988341704275415, −2.348505276236219, −2.012768539611614, −1.067284445760304, −0.7197690430725214,
0.7197690430725214, 1.067284445760304, 2.012768539611614, 2.348505276236219, 2.988341704275415, 3.627627063877335, 3.954915430144945, 4.395706144017710, 5.058943200332452, 5.342724390704455, 6.149809767351052, 6.558576087818673, 6.977595079376814, 7.592179509858635, 7.871990763085844, 8.431835600256164, 9.052331108005281, 9.210382794628510, 9.846505951754306, 10.51918639932818, 10.92851597745078, 11.24342331722858, 11.78551827499263, 12.18490542426713, 12.63397935572917