L(s) = 1 | − 2·7-s − 11-s − 13-s − 17-s + 4·19-s + 23-s + 5·29-s + 31-s − 6·37-s − 7·43-s + 7·47-s − 3·49-s − 12·53-s + 4·59-s + 10·61-s + 4·67-s − 12·71-s − 6·73-s + 2·77-s + 15·79-s + 2·83-s + 12·89-s + 2·91-s − 10·97-s + 15·101-s + 14·103-s + 18·107-s + ⋯ |
L(s) = 1 | − 0.755·7-s − 0.301·11-s − 0.277·13-s − 0.242·17-s + 0.917·19-s + 0.208·23-s + 0.928·29-s + 0.179·31-s − 0.986·37-s − 1.06·43-s + 1.02·47-s − 3/7·49-s − 1.64·53-s + 0.520·59-s + 1.28·61-s + 0.488·67-s − 1.42·71-s − 0.702·73-s + 0.227·77-s + 1.68·79-s + 0.219·83-s + 1.27·89-s + 0.209·91-s − 1.01·97-s + 1.49·101-s + 1.37·103-s + 1.74·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.480881895\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.480881895\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.184622414576606606317287853770, −7.38748725149846239732154601567, −6.76495726913731380199515593188, −6.09788577130007395789882661902, −5.22644316417081378559806102544, −4.61211048766057560995193968160, −3.50478602576255188217533958541, −2.97046151281418265357600520916, −1.92970764474191076466338159375, −0.64123253528122429969177517197,
0.64123253528122429969177517197, 1.92970764474191076466338159375, 2.97046151281418265357600520916, 3.50478602576255188217533958541, 4.61211048766057560995193968160, 5.22644316417081378559806102544, 6.09788577130007395789882661902, 6.76495726913731380199515593188, 7.38748725149846239732154601567, 8.184622414576606606317287853770