| L(s) = 1 | − 2-s + 4-s − 8-s − 3·9-s − 2·11-s + 16-s − 7·17-s + 3·18-s + 2·22-s − 3·23-s + 6·29-s + 7·31-s − 32-s + 7·34-s − 3·36-s + 4·37-s + 7·41-s + 8·43-s − 2·44-s + 3·46-s − 7·47-s − 4·53-s − 6·58-s + 14·59-s + 14·61-s − 7·62-s + 64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 9-s − 0.603·11-s + 1/4·16-s − 1.69·17-s + 0.707·18-s + 0.426·22-s − 0.625·23-s + 1.11·29-s + 1.25·31-s − 0.176·32-s + 1.20·34-s − 1/2·36-s + 0.657·37-s + 1.09·41-s + 1.21·43-s − 0.301·44-s + 0.442·46-s − 1.02·47-s − 0.549·53-s − 0.787·58-s + 1.82·59-s + 1.79·61-s − 0.889·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8723378954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8723378954\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−8.762484884389541839139246327012, −8.351818921227748995788738370950, −7.60391804514523541201006005940, −6.58340523729974343387448028554, −6.07855296739393657632743733560, −5.05176429892797620868408295477, −4.12750588190300947653015345104, −2.79879479590849346841634277415, −2.26337827706355780177290987463, −0.63103404345823596261328848038,
0.63103404345823596261328848038, 2.26337827706355780177290987463, 2.79879479590849346841634277415, 4.12750588190300947653015345104, 5.05176429892797620868408295477, 6.07855296739393657632743733560, 6.58340523729974343387448028554, 7.60391804514523541201006005940, 8.351818921227748995788738370950, 8.762484884389541839139246327012
