| L(s) = 1 | − 1.50·2-s + 3.04·3-s + 0.256·4-s + 3.82·5-s − 4.56·6-s + 2.61·8-s + 6.24·9-s − 5.74·10-s − 0.542·11-s + 0.779·12-s + 1.21·13-s + 11.6·15-s − 4.44·16-s + 3.66·17-s − 9.38·18-s − 5.00·19-s + 0.980·20-s + 0.814·22-s − 23-s + 7.96·24-s + 9.62·25-s − 1.82·26-s + 9.86·27-s + 3.72·29-s − 17.4·30-s − 9.04·31-s + 1.44·32-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 1.75·3-s + 0.128·4-s + 1.71·5-s − 1.86·6-s + 0.925·8-s + 2.08·9-s − 1.81·10-s − 0.163·11-s + 0.225·12-s + 0.337·13-s + 3.00·15-s − 1.11·16-s + 0.888·17-s − 2.21·18-s − 1.14·19-s + 0.219·20-s + 0.173·22-s − 0.208·23-s + 1.62·24-s + 1.92·25-s − 0.358·26-s + 1.89·27-s + 0.692·29-s − 3.18·30-s − 1.62·31-s + 0.254·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.240319654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.240319654\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 1.50T + 2T^{2} \) |
| 3 | \( 1 - 3.04T + 3T^{2} \) |
| 5 | \( 1 - 3.82T + 5T^{2} \) |
| 11 | \( 1 + 0.542T + 11T^{2} \) |
| 13 | \( 1 - 1.21T + 13T^{2} \) |
| 17 | \( 1 - 3.66T + 17T^{2} \) |
| 19 | \( 1 + 5.00T + 19T^{2} \) |
| 29 | \( 1 - 3.72T + 29T^{2} \) |
| 31 | \( 1 + 9.04T + 31T^{2} \) |
| 37 | \( 1 + 9.08T + 37T^{2} \) |
| 41 | \( 1 - 4.86T + 41T^{2} \) |
| 43 | \( 1 - 3.64T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 - 7.48T + 53T^{2} \) |
| 59 | \( 1 + 5.18T + 59T^{2} \) |
| 61 | \( 1 + 3.26T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 + 5.29T + 71T^{2} \) |
| 73 | \( 1 + 8.19T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 + 7.64T + 83T^{2} \) |
| 89 | \( 1 - 4.74T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−9.556486017774339739071685111575, −9.015958603661334844643120486179, −8.529456297767110835124553120759, −7.69191262514044781524298334249, −6.83968911116704990607133589252, −5.68947140093051763698108206999, −4.47288681290867943778606915594, −3.23916169548805405186439517096, −2.11000952100360740886281182478, −1.51694936648245565577982886385,
1.51694936648245565577982886385, 2.11000952100360740886281182478, 3.23916169548805405186439517096, 4.47288681290867943778606915594, 5.68947140093051763698108206999, 6.83968911116704990607133589252, 7.69191262514044781524298334249, 8.529456297767110835124553120759, 9.015958603661334844643120486179, 9.556486017774339739071685111575
