| L(s) = 1 | + 1.78·3-s + 1.75·7-s + 0.192·9-s + 4.77·11-s − 1.72·13-s + 7.81·17-s + 2.43·19-s + 3.13·21-s − 23-s − 5.01·27-s + 7.86·29-s − 6.14·31-s + 8.53·33-s + 6.83·37-s − 3.09·39-s − 2.50·41-s − 3.26·43-s + 8.46·47-s − 3.91·49-s + 13.9·51-s + 2.76·53-s + 4.35·57-s − 1.91·59-s − 3.50·61-s + 0.337·63-s − 12.7·67-s − 1.78·69-s + ⋯ |
| L(s) = 1 | + 1.03·3-s + 0.663·7-s + 0.0640·9-s + 1.43·11-s − 0.479·13-s + 1.89·17-s + 0.559·19-s + 0.684·21-s − 0.208·23-s − 0.965·27-s + 1.46·29-s − 1.10·31-s + 1.48·33-s + 1.12·37-s − 0.494·39-s − 0.391·41-s − 0.498·43-s + 1.23·47-s − 0.559·49-s + 1.95·51-s + 0.379·53-s + 0.577·57-s − 0.249·59-s − 0.448·61-s + 0.0424·63-s − 1.55·67-s − 0.215·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.995341657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.995341657\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 1.78T + 3T^{2} \) |
| 7 | \( 1 - 1.75T + 7T^{2} \) |
| 11 | \( 1 - 4.77T + 11T^{2} \) |
| 13 | \( 1 + 1.72T + 13T^{2} \) |
| 17 | \( 1 - 7.81T + 17T^{2} \) |
| 19 | \( 1 - 2.43T + 19T^{2} \) |
| 29 | \( 1 - 7.86T + 29T^{2} \) |
| 31 | \( 1 + 6.14T + 31T^{2} \) |
| 37 | \( 1 - 6.83T + 37T^{2} \) |
| 41 | \( 1 + 2.50T + 41T^{2} \) |
| 43 | \( 1 + 3.26T + 43T^{2} \) |
| 47 | \( 1 - 8.46T + 47T^{2} \) |
| 53 | \( 1 - 2.76T + 53T^{2} \) |
| 59 | \( 1 + 1.91T + 59T^{2} \) |
| 61 | \( 1 + 3.50T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + 0.0111T + 73T^{2} \) |
| 79 | \( 1 - 16.6T + 79T^{2} \) |
| 83 | \( 1 + 2.64T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
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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
−7.75351699523983063408995361088, −7.31834128510286464086629479705, −6.38092415656938221332597115706, −5.64690566495333872335052082677, −4.91054480327030219831361563422, −4.02245051226088887144274720491, −3.41282794943977006413364058844, −2.72628991232437071501954548967, −1.73428420247835297824018010066, −0.989906253540845912576349672566,
0.989906253540845912576349672566, 1.73428420247835297824018010066, 2.72628991232437071501954548967, 3.41282794943977006413364058844, 4.02245051226088887144274720491, 4.91054480327030219831361563422, 5.64690566495333872335052082677, 6.38092415656938221332597115706, 7.31834128510286464086629479705, 7.75351699523983063408995361088
