| L(s) = 1 | + 4·5-s − 4·11-s − 4·13-s + 4·19-s + 11·25-s − 2·29-s + 8·31-s − 6·37-s − 4·43-s + 8·47-s + 10·53-s − 16·55-s − 4·59-s + 4·61-s − 16·65-s − 4·67-s + 8·71-s + 16·73-s + 8·79-s + 12·83-s + 8·89-s + 16·95-s − 8·97-s + 4·101-s − 8·103-s + 4·107-s − 14·109-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.20·11-s − 1.10·13-s + 0.917·19-s + 11/5·25-s − 0.371·29-s + 1.43·31-s − 0.986·37-s − 0.609·43-s + 1.16·47-s + 1.37·53-s − 2.15·55-s − 0.520·59-s + 0.512·61-s − 1.98·65-s − 0.488·67-s + 0.949·71-s + 1.87·73-s + 0.900·79-s + 1.31·83-s + 0.847·89-s + 1.64·95-s − 0.812·97-s + 0.398·101-s − 0.788·103-s + 0.386·107-s − 1.34·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.656413816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.656413816\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.916680638172122349209180978402, −7.15626924228060006589866403814, −6.53223269447526142619685193559, −5.64836882923047051143323621301, −5.26001411946587964391418777104, −4.69336426412013614248722189397, −3.32288223921369618581771511826, −2.48298505359813833416816457120, −2.06015196877250864953474557133, −0.818745509716799952266967219226,
0.818745509716799952266967219226, 2.06015196877250864953474557133, 2.48298505359813833416816457120, 3.32288223921369618581771511826, 4.69336426412013614248722189397, 5.26001411946587964391418777104, 5.64836882923047051143323621301, 6.53223269447526142619685193559, 7.15626924228060006589866403814, 7.916680638172122349209180978402
