| L(s) = 1 | + 2·5-s + 11-s + 2·13-s − 2·19-s − 25-s + 6·29-s − 4·31-s − 2·37-s + 8·41-s − 12·43-s + 12·47-s − 2·53-s + 2·55-s − 10·59-s − 10·61-s + 4·65-s + 12·67-s − 4·71-s − 12·73-s + 18·83-s − 4·95-s + 12·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.301·11-s + 0.554·13-s − 0.458·19-s − 1/5·25-s + 1.11·29-s − 0.718·31-s − 0.328·37-s + 1.24·41-s − 1.82·43-s + 1.75·47-s − 0.274·53-s + 0.269·55-s − 1.30·59-s − 1.28·61-s + 0.496·65-s + 1.46·67-s − 0.474·71-s − 1.40·73-s + 1.97·83-s − 0.410·95-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.981694345\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.981694345\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−12.64004686629171, −12.31081101758655, −11.70601100209259, −11.31065473014081, −10.68545666016524, −10.42496523386524, −9.948706789396547, −9.427767738102927, −8.881949100453580, −8.777363366819091, −7.984787011415179, −7.603167747566256, −6.998685447732904, −6.438462860148448, −6.060756771493558, −5.780178282735060, −4.983162217632253, −4.702090437854868, −3.941371974334854, −3.551451659285603, −2.841051124287552, −2.323771150081617, −1.709997445905531, −1.257396665443184, −0.4539736169653334,
0.4539736169653334, 1.257396665443184, 1.709997445905531, 2.323771150081617, 2.841051124287552, 3.551451659285603, 3.941371974334854, 4.702090437854868, 4.983162217632253, 5.780178282735060, 6.060756771493558, 6.438462860148448, 6.998685447732904, 7.603167747566256, 7.984787011415179, 8.777363366819091, 8.881949100453580, 9.427767738102927, 9.948706789396547, 10.42496523386524, 10.68545666016524, 11.31065473014081, 11.70601100209259, 12.31081101758655, 12.64004686629171
