| L(s) = 1 | − 2-s + 4-s − 1.41·5-s − 8-s + 1.41·10-s − 2.24·11-s + 3·13-s + 16-s − 4.41·17-s + 4.58·19-s − 1.41·20-s + 2.24·22-s + 23-s − 2.99·25-s − 3·26-s + 5.24·29-s − 1.24·31-s − 32-s + 4.41·34-s − 10.4·37-s − 4.58·38-s + 1.41·40-s + 2.82·41-s + 3.24·43-s − 2.24·44-s − 46-s + 7.07·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.632·5-s − 0.353·8-s + 0.447·10-s − 0.676·11-s + 0.832·13-s + 0.250·16-s − 1.07·17-s + 1.05·19-s − 0.316·20-s + 0.478·22-s + 0.208·23-s − 0.599·25-s − 0.588·26-s + 0.973·29-s − 0.223·31-s − 0.176·32-s + 0.757·34-s − 1.72·37-s − 0.743·38-s + 0.223·40-s + 0.441·41-s + 0.494·43-s − 0.338·44-s − 0.147·46-s + 1.03·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9551862906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9551862906\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 1.41T + 5T^{2} \) |
| 11 | \( 1 + 2.24T + 11T^{2} \) |
| 13 | \( 1 - 3T + 13T^{2} \) |
| 17 | \( 1 + 4.41T + 17T^{2} \) |
| 19 | \( 1 - 4.58T + 19T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 29 | \( 1 - 5.24T + 29T^{2} \) |
| 31 | \( 1 + 1.24T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 - 2.82T + 41T^{2} \) |
| 43 | \( 1 - 3.24T + 43T^{2} \) |
| 47 | \( 1 - 7.07T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 0.171T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 - 5.48T + 71T^{2} \) |
| 73 | \( 1 - 9.89T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 17.3T + 83T^{2} \) |
| 89 | \( 1 - 1.92T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−8.699149797925248000354787794766, −8.237286489820970190517328992341, −7.42275824885054975414024466659, −6.80604750297983364393619751308, −5.87216411657332347913594720485, −4.99186653630415690985044447152, −3.94031690876713820832492974536, −3.08630207941805231435906236619, −1.99755420503591486236034862015, −0.67186975612061848528006318728,
0.67186975612061848528006318728, 1.99755420503591486236034862015, 3.08630207941805231435906236619, 3.94031690876713820832492974536, 4.99186653630415690985044447152, 5.87216411657332347913594720485, 6.80604750297983364393619751308, 7.42275824885054975414024466659, 8.237286489820970190517328992341, 8.699149797925248000354787794766
