| L(s) = 1 | + 7-s + 5·11-s + 3·13-s − 17-s + 6·19-s + 6·23-s + 9·29-s − 4·31-s − 2·37-s + 4·41-s − 10·43-s − 47-s + 49-s + 4·53-s + 8·59-s − 8·61-s − 12·67-s − 8·71-s − 2·73-s + 5·77-s + 13·79-s − 4·83-s − 4·89-s + 3·91-s + 13·97-s − 6·101-s − 19·103-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.50·11-s + 0.832·13-s − 0.242·17-s + 1.37·19-s + 1.25·23-s + 1.67·29-s − 0.718·31-s − 0.328·37-s + 0.624·41-s − 1.52·43-s − 0.145·47-s + 1/7·49-s + 0.549·53-s + 1.04·59-s − 1.02·61-s − 1.46·67-s − 0.949·71-s − 0.234·73-s + 0.569·77-s + 1.46·79-s − 0.439·83-s − 0.423·89-s + 0.314·91-s + 1.31·97-s − 0.597·101-s − 1.87·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.778847224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.778847224\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−8.112266990729140719928976691570, −7.19188147123175063389848783861, −6.71283941834008360705421541340, −5.96986775775042332086987702841, −5.14013781533503607240800407374, −4.42298800129827627767671594970, −3.58882467704305349487903097087, −2.91993373267792540868956023873, −1.57823619004854753987046866680, −0.984574808565519017599769408630,
0.984574808565519017599769408630, 1.57823619004854753987046866680, 2.91993373267792540868956023873, 3.58882467704305349487903097087, 4.42298800129827627767671594970, 5.14013781533503607240800407374, 5.96986775775042332086987702841, 6.71283941834008360705421541340, 7.19188147123175063389848783861, 8.112266990729140719928976691570
