L(s) = 1 | + 4·5-s − 3·13-s − 4·17-s + 7·19-s + 4·23-s + 11·25-s − 8·29-s − 5·31-s + 3·37-s + 8·41-s + 11·43-s + 4·47-s + 4·53-s + 12·59-s − 2·61-s − 12·65-s − 3·67-s + 12·71-s + 73-s + 79-s + 12·83-s − 16·85-s + 8·89-s + 28·95-s − 2·97-s + 3·103-s − 12·107-s + ⋯ |
L(s) = 1 | + 1.78·5-s − 0.832·13-s − 0.970·17-s + 1.60·19-s + 0.834·23-s + 11/5·25-s − 1.48·29-s − 0.898·31-s + 0.493·37-s + 1.24·41-s + 1.67·43-s + 0.583·47-s + 0.549·53-s + 1.56·59-s − 0.256·61-s − 1.48·65-s − 0.366·67-s + 1.42·71-s + 0.117·73-s + 0.112·79-s + 1.31·83-s − 1.73·85-s + 0.847·89-s + 2.87·95-s − 0.203·97-s + 0.295·103-s − 1.16·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.717356781\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.717356781\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.953878870337658147748039146837, −7.62169342394056432219755362912, −7.10795510504113439018737765605, −6.23748725310251648874990762582, −5.46198590764031912402717426184, −5.11412492130608758887594001449, −3.91997383953374330705165521335, −2.67482201641444518746371738530, −2.15957796084926510385840541081, −1.00963338520985869910017808867,
1.00963338520985869910017808867, 2.15957796084926510385840541081, 2.67482201641444518746371738530, 3.91997383953374330705165521335, 5.11412492130608758887594001449, 5.46198590764031912402717426184, 6.23748725310251648874990762582, 7.10795510504113439018737765605, 7.62169342394056432219755362912, 8.953878870337658147748039146837