| L(s) = 1 | − 2-s − 4-s + 2.85·5-s − 7-s + 3·8-s − 2.85·10-s + 1.23·13-s + 14-s − 16-s + 7.85·17-s − 2.61·19-s − 2.85·20-s + 3.09·23-s + 3.14·25-s − 1.23·26-s + 28-s − 2·29-s − 7.32·31-s − 5·32-s − 7.85·34-s − 2.85·35-s − 12.0·37-s + 2.61·38-s + 8.56·40-s − 10.8·41-s + 1.23·43-s − 3.09·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.5·4-s + 1.27·5-s − 0.377·7-s + 1.06·8-s − 0.902·10-s + 0.342·13-s + 0.267·14-s − 0.250·16-s + 1.90·17-s − 0.600·19-s − 0.638·20-s + 0.644·23-s + 0.629·25-s − 0.242·26-s + 0.188·28-s − 0.371·29-s − 1.31·31-s − 0.883·32-s − 1.34·34-s − 0.482·35-s − 1.98·37-s + 0.424·38-s + 1.35·40-s − 1.69·41-s + 0.188·43-s − 0.455·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + 2T^{2} \) |
| 5 | \( 1 - 2.85T + 5T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 - 7.85T + 17T^{2} \) |
| 19 | \( 1 + 2.61T + 19T^{2} \) |
| 23 | \( 1 - 3.09T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 7.32T + 31T^{2} \) |
| 37 | \( 1 + 12.0T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 - 1.23T + 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 + 6.76T + 59T^{2} \) |
| 61 | \( 1 + 8.94T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 - 5.23T + 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 - 2.94T + 83T^{2} \) |
| 89 | \( 1 + 1.09T + 89T^{2} \) |
| 97 | \( 1 + 3.52T + 97T^{2} \) |
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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
−7.66340059004799438838515002534, −6.88603948178923700006979658522, −6.14632144519866587344601277051, −5.34707741832851384457919340182, −5.02293874384626147573678822029, −3.74501388039750696524107084598, −3.17463509154787482144334220198, −1.83374285155447229927655523625, −1.36174737978857535738788788923, 0,
1.36174737978857535738788788923, 1.83374285155447229927655523625, 3.17463509154787482144334220198, 3.74501388039750696524107084598, 5.02293874384626147573678822029, 5.34707741832851384457919340182, 6.14632144519866587344601277051, 6.88603948178923700006979658522, 7.66340059004799438838515002534
