| L(s) = 1 | + 1.78·3-s + 4.18·5-s + 7-s + 0.189·9-s + 4.50·11-s + 4.84·13-s + 7.46·15-s + 5.13·17-s − 2.12·19-s + 1.78·21-s − 7.11·23-s + 12.4·25-s − 5.01·27-s + 5.43·29-s − 0.831·31-s + 8.04·33-s + 4.18·35-s − 7.98·37-s + 8.65·39-s − 2.22·41-s − 2.28·43-s + 0.790·45-s + 7.83·47-s + 49-s + 9.17·51-s − 7.90·53-s + 18.8·55-s + ⋯ |
| L(s) = 1 | + 1.03·3-s + 1.87·5-s + 0.377·7-s + 0.0630·9-s + 1.35·11-s + 1.34·13-s + 1.92·15-s + 1.24·17-s − 0.488·19-s + 0.389·21-s − 1.48·23-s + 2.49·25-s − 0.966·27-s + 1.01·29-s − 0.149·31-s + 1.40·33-s + 0.706·35-s − 1.31·37-s + 1.38·39-s − 0.347·41-s − 0.348·43-s + 0.117·45-s + 1.14·47-s + 0.142·49-s + 1.28·51-s − 1.08·53-s + 2.53·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.388606776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.388606776\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 - 1.78T + 3T^{2} \) |
| 5 | \( 1 - 4.18T + 5T^{2} \) |
| 11 | \( 1 - 4.50T + 11T^{2} \) |
| 13 | \( 1 - 4.84T + 13T^{2} \) |
| 17 | \( 1 - 5.13T + 17T^{2} \) |
| 19 | \( 1 + 2.12T + 19T^{2} \) |
| 23 | \( 1 + 7.11T + 23T^{2} \) |
| 29 | \( 1 - 5.43T + 29T^{2} \) |
| 31 | \( 1 + 0.831T + 31T^{2} \) |
| 37 | \( 1 + 7.98T + 37T^{2} \) |
| 41 | \( 1 + 2.22T + 41T^{2} \) |
| 43 | \( 1 + 2.28T + 43T^{2} \) |
| 47 | \( 1 - 7.83T + 47T^{2} \) |
| 53 | \( 1 + 7.90T + 53T^{2} \) |
| 59 | \( 1 - 2.62T + 59T^{2} \) |
| 61 | \( 1 - 2.33T + 61T^{2} \) |
| 67 | \( 1 + 8.16T + 67T^{2} \) |
| 71 | \( 1 + 6.04T + 71T^{2} \) |
| 73 | \( 1 + 7.67T + 73T^{2} \) |
| 79 | \( 1 + 1.90T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 + 2.49T + 89T^{2} \) |
| 97 | \( 1 + 1.98T + 97T^{2} \) |
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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
−8.209205253568209120392898176577, −7.22248731922738012751199294186, −6.26340872628893219788280560546, −6.03429736848862887040056862120, −5.27660904143873465723130347794, −4.15828926575123071394152752176, −3.45622517998227019599589051130, −2.64444835546814729846663915681, −1.67671400631649522553928914543, −1.37033412266203473872607281353,
1.37033412266203473872607281353, 1.67671400631649522553928914543, 2.64444835546814729846663915681, 3.45622517998227019599589051130, 4.15828926575123071394152752176, 5.27660904143873465723130347794, 6.03429736848862887040056862120, 6.26340872628893219788280560546, 7.22248731922738012751199294186, 8.209205253568209120392898176577
