L(s) = 1 | − 0.160·2-s − 1.99·3-s − 1.97·4-s + 0.944·5-s + 0.318·6-s − 2.45·7-s + 0.636·8-s + 0.960·9-s − 0.151·10-s + 3.92·12-s + 1.49·13-s + 0.393·14-s − 1.87·15-s + 3.84·16-s + 4.43·17-s − 0.153·18-s + 8.13·19-s − 1.86·20-s + 4.89·21-s + 2.72·23-s − 1.26·24-s − 4.10·25-s − 0.238·26-s + 4.05·27-s + 4.85·28-s + 0.129·29-s + 0.301·30-s + ⋯ |
L(s) = 1 | − 0.113·2-s − 1.14·3-s − 0.987·4-s + 0.422·5-s + 0.130·6-s − 0.929·7-s + 0.225·8-s + 0.320·9-s − 0.0478·10-s + 1.13·12-s + 0.413·13-s + 0.105·14-s − 0.485·15-s + 0.961·16-s + 1.07·17-s − 0.0362·18-s + 1.86·19-s − 0.416·20-s + 1.06·21-s + 0.567·23-s − 0.258·24-s − 0.821·25-s − 0.0468·26-s + 0.781·27-s + 0.917·28-s + 0.0240·29-s + 0.0549·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8589847057\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8589847057\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 61 | \( 1 + T \) |
good | 2 | \( 1 + 0.160T + 2T^{2} \) |
| 3 | \( 1 + 1.99T + 3T^{2} \) |
| 5 | \( 1 - 0.944T + 5T^{2} \) |
| 7 | \( 1 + 2.45T + 7T^{2} \) |
| 13 | \( 1 - 1.49T + 13T^{2} \) |
| 17 | \( 1 - 4.43T + 17T^{2} \) |
| 19 | \( 1 - 8.13T + 19T^{2} \) |
| 23 | \( 1 - 2.72T + 23T^{2} \) |
| 29 | \( 1 - 0.129T + 29T^{2} \) |
| 31 | \( 1 + 6.71T + 31T^{2} \) |
| 37 | \( 1 - 9.55T + 37T^{2} \) |
| 41 | \( 1 + 2.01T + 41T^{2} \) |
| 43 | \( 1 - 5.42T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 4.25T + 53T^{2} \) |
| 59 | \( 1 + 1.18T + 59T^{2} \) |
| 67 | \( 1 + 5.70T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 6.56T + 73T^{2} \) |
| 79 | \( 1 + 1.69T + 79T^{2} \) |
| 83 | \( 1 + 0.566T + 83T^{2} \) |
| 89 | \( 1 + 2.01T + 89T^{2} \) |
| 97 | \( 1 - 9.22T + 97T^{2} \) |
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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
−7.66289845460332188972819686501, −7.33775438956177304430424312902, −6.11298078375985011735261853337, −5.81624047094459609531393421418, −5.31814210712309064258133238557, −4.48908493892374349995863977759, −3.55526013409187177346696992089, −2.93840859137259524512260263217, −1.31404998159399527843253248164, −0.57861472676279069349383060772,
0.57861472676279069349383060772, 1.31404998159399527843253248164, 2.93840859137259524512260263217, 3.55526013409187177346696992089, 4.48908493892374349995863977759, 5.31814210712309064258133238557, 5.81624047094459609531393421418, 6.11298078375985011735261853337, 7.33775438956177304430424312902, 7.66289845460332188972819686501