| L(s) = 1 | − 5-s + 7-s + 6·11-s − 13-s + 8·17-s + 3·19-s − 9·23-s − 4·25-s + 9·29-s − 31-s − 35-s − 8·37-s − 2·41-s + 9·43-s + 3·47-s + 49-s + 5·53-s − 6·55-s − 8·59-s − 10·61-s + 65-s − 10·67-s + 12·71-s + 3·73-s + 6·77-s − 7·79-s + 15·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 1.80·11-s − 0.277·13-s + 1.94·17-s + 0.688·19-s − 1.87·23-s − 4/5·25-s + 1.67·29-s − 0.179·31-s − 0.169·35-s − 1.31·37-s − 0.312·41-s + 1.37·43-s + 0.437·47-s + 1/7·49-s + 0.686·53-s − 0.809·55-s − 1.04·59-s − 1.28·61-s + 0.124·65-s − 1.22·67-s + 1.42·71-s + 0.351·73-s + 0.683·77-s − 0.787·79-s + 1.64·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.102043545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.102043545\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 - 15 T + p T^{2} \) |
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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
−8.588943294913916144303770226519, −7.81017917770960962493477778363, −7.34720187422893546120672011284, −6.29947170772173464283060055039, −5.73838646791182021882632100284, −4.69034388954423580716740084793, −3.88102117197123216567783894059, −3.28850193522786672972611735122, −1.87027540489607850169666877892, −0.927402921974082899230984722556,
0.927402921974082899230984722556, 1.87027540489607850169666877892, 3.28850193522786672972611735122, 3.88102117197123216567783894059, 4.69034388954423580716740084793, 5.73838646791182021882632100284, 6.29947170772173464283060055039, 7.34720187422893546120672011284, 7.81017917770960962493477778363, 8.588943294913916144303770226519
