L(s) = 1 | − 0.271·2-s + 0.319·3-s − 1.92·4-s − 0.0867·6-s − 3.38·7-s + 1.06·8-s − 2.89·9-s + 1.75·11-s − 0.615·12-s + 0.917·14-s + 3.56·16-s − 1.95·17-s + 0.786·18-s − 7.13·19-s − 1.08·21-s − 0.475·22-s − 7.61·23-s + 0.340·24-s − 1.88·27-s + 6.51·28-s + 3.98·29-s + 4.86·31-s − 3.09·32-s + 0.559·33-s + 0.530·34-s + 5.58·36-s − 10.5·37-s + ⋯ |
L(s) = 1 | − 0.191·2-s + 0.184·3-s − 0.963·4-s − 0.0354·6-s − 1.27·7-s + 0.376·8-s − 0.965·9-s + 0.528·11-s − 0.177·12-s + 0.245·14-s + 0.890·16-s − 0.473·17-s + 0.185·18-s − 1.63·19-s − 0.235·21-s − 0.101·22-s − 1.58·23-s + 0.0695·24-s − 0.362·27-s + 1.23·28-s + 0.739·29-s + 0.874·31-s − 0.547·32-s + 0.0974·33-s + 0.0909·34-s + 0.930·36-s − 1.72·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4567371224\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4567371224\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 0.271T + 2T^{2} \) |
| 3 | \( 1 - 0.319T + 3T^{2} \) |
| 7 | \( 1 + 3.38T + 7T^{2} \) |
| 11 | \( 1 - 1.75T + 11T^{2} \) |
| 17 | \( 1 + 1.95T + 17T^{2} \) |
| 19 | \( 1 + 7.13T + 19T^{2} \) |
| 23 | \( 1 + 7.61T + 23T^{2} \) |
| 29 | \( 1 - 3.98T + 29T^{2} \) |
| 31 | \( 1 - 4.86T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 - 0.911T + 41T^{2} \) |
| 43 | \( 1 + 4.58T + 43T^{2} \) |
| 47 | \( 1 - 8.58T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 - 3.82T + 59T^{2} \) |
| 61 | \( 1 - 7.98T + 61T^{2} \) |
| 67 | \( 1 + 0.472T + 67T^{2} \) |
| 71 | \( 1 + 5.50T + 71T^{2} \) |
| 73 | \( 1 + 2.93T + 73T^{2} \) |
| 79 | \( 1 - 4.09T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + 3.85T + 89T^{2} \) |
| 97 | \( 1 - 6.95T + 97T^{2} \) |
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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
−8.535581146882649443896242173629, −7.985648263483693282076992025888, −6.73515667479236811516395213133, −6.29205728491981516284882061985, −5.52687365358230507614024275048, −4.46863513839017758333313954818, −3.83947116363504224524880063012, −3.05346768738306111773972340242, −1.99494622340803464077276864585, −0.37305578049020378778633239667,
0.37305578049020378778633239667, 1.99494622340803464077276864585, 3.05346768738306111773972340242, 3.83947116363504224524880063012, 4.46863513839017758333313954818, 5.52687365358230507614024275048, 6.29205728491981516284882061985, 6.73515667479236811516395213133, 7.985648263483693282076992025888, 8.535581146882649443896242173629