| L(s) = 1 | − 2.23·2-s + 3-s + 3.00·4-s − 2.23·6-s + 1.23·7-s − 2.23·8-s + 9-s + 4·11-s + 3.00·12-s − 4.47·13-s − 2.76·14-s − 0.999·16-s + 7.23·17-s − 2.23·18-s + 2.76·19-s + 1.23·21-s − 8.94·22-s − 23-s − 2.23·24-s + 10.0·26-s + 27-s + 3.70·28-s − 4.47·29-s + 2.47·31-s + 6.70·32-s + 4·33-s − 16.1·34-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 0.577·3-s + 1.50·4-s − 0.912·6-s + 0.467·7-s − 0.790·8-s + 0.333·9-s + 1.20·11-s + 0.866·12-s − 1.24·13-s − 0.738·14-s − 0.249·16-s + 1.75·17-s − 0.527·18-s + 0.634·19-s + 0.269·21-s − 1.90·22-s − 0.208·23-s − 0.456·24-s + 1.96·26-s + 0.192·27-s + 0.700·28-s − 0.830·29-s + 0.444·31-s + 1.18·32-s + 0.696·33-s − 2.77·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.136750886\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.136750886\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 7.23T + 17T^{2} \) |
| 19 | \( 1 - 2.76T + 19T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 - 2.47T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 - 6.94T + 41T^{2} \) |
| 43 | \( 1 + 7.70T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 0.763T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 + 4.47T + 61T^{2} \) |
| 67 | \( 1 + 5.23T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 3.70T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 3.23T + 89T^{2} \) |
| 97 | \( 1 - 0.472T + 97T^{2} \) |
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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
−9.423601154842827416136640203335, −8.609865643687584145814140494184, −7.71727870379861003800066083423, −7.50036789352519607150077351589, −6.51697029582375358696865609159, −5.33998168833763295369294349275, −4.19721173499108813684478401692, −3.02677733248705294704741263461, −1.88869475893590461747925331408, −0.965181331712455148186417555623,
0.965181331712455148186417555623, 1.88869475893590461747925331408, 3.02677733248705294704741263461, 4.19721173499108813684478401692, 5.33998168833763295369294349275, 6.51697029582375358696865609159, 7.50036789352519607150077351589, 7.71727870379861003800066083423, 8.609865643687584145814140494184, 9.423601154842827416136640203335
