L(s) = 1 | + 3-s − 5-s − 2·7-s + 9-s + 13-s − 15-s + 17-s + 4·19-s − 2·21-s + 8·23-s + 25-s + 27-s + 2·29-s + 2·31-s + 2·35-s + 4·37-s + 39-s − 6·41-s + 12·43-s − 45-s − 6·47-s − 3·49-s + 51-s + 6·53-s + 4·57-s − 2·59-s − 2·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.277·13-s − 0.258·15-s + 0.242·17-s + 0.917·19-s − 0.436·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.359·31-s + 0.338·35-s + 0.657·37-s + 0.160·39-s − 0.937·41-s + 1.82·43-s − 0.149·45-s − 0.875·47-s − 3/7·49-s + 0.140·51-s + 0.824·53-s + 0.529·57-s − 0.260·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.651378685\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.651378685\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−15.37459384102004, −14.78178571870702, −14.21866579170545, −13.72373357642452, −13.03658378228373, −12.83379373219606, −12.09500811177173, −11.54993019827311, −10.98858190620828, −10.32945151771798, −9.803812768510371, −9.145169171271912, −8.883675952449925, −8.014442419143513, −7.637600501748073, −6.912335593474231, −6.516087999032995, −5.676834914818590, −5.006141164399350, −4.334823271832925, −3.560601889517154, −3.084417190903902, −2.550221807511151, −1.381922928316845, −0.6649683820279837,
0.6649683820279837, 1.381922928316845, 2.550221807511151, 3.084417190903902, 3.560601889517154, 4.334823271832925, 5.006141164399350, 5.676834914818590, 6.516087999032995, 6.912335593474231, 7.637600501748073, 8.014442419143513, 8.883675952449925, 9.145169171271912, 9.803812768510371, 10.32945151771798, 10.98858190620828, 11.54993019827311, 12.09500811177173, 12.83379373219606, 13.03658378228373, 13.72373357642452, 14.21866579170545, 14.78178571870702, 15.37459384102004