| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 8-s + 3·9-s + 2·12-s + 16-s − 3·18-s − 2·24-s + 4·27-s − 29-s − 32-s + 3·36-s − 2·43-s − 2·47-s + 2·48-s + 49-s − 4·54-s + 58-s + 64-s − 3·72-s + 5·81-s + 2·86-s − 2·87-s + 2·94-s − 2·96-s + ⋯ |
| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 8-s + 3·9-s + 2·12-s + 16-s − 3·18-s − 2·24-s + 4·27-s − 29-s − 32-s + 3·36-s − 2·43-s − 2·47-s + 2·48-s + 49-s − 4·54-s + 58-s + 64-s − 3·72-s + 5·81-s + 2·86-s − 2·87-s + 2·94-s − 2·96-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.688603262\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.688603262\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( ( 1 - T )^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 + T )^{2} \) |
| 47 | \( ( 1 + T )^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−8.739263448443951963768192946869, −8.428944285996751718909818997348, −7.65204245929723613468130482820, −7.13273108166356499472271347594, −6.33403057552372579969061849871, −4.93550520979361483113148670468, −3.74622317954035227069607203779, −3.14582559267984043906849720568, −2.21581852805476656076166455263, −1.48085941576244790466121624626,
1.48085941576244790466121624626, 2.21581852805476656076166455263, 3.14582559267984043906849720568, 3.74622317954035227069607203779, 4.93550520979361483113148670468, 6.33403057552372579969061849871, 7.13273108166356499472271347594, 7.65204245929723613468130482820, 8.428944285996751718909818997348, 8.739263448443951963768192946869
