| L(s) = 1 | + 3-s + 5-s − 7-s + 15-s + 2·17-s + 19-s − 21-s − 27-s + 29-s − 35-s − 41-s + 2·51-s + 53-s + 57-s − 59-s + 2·71-s − 79-s − 81-s + 2·85-s + 87-s + 95-s − 105-s + 107-s − 2·119-s + ⋯ |
| L(s) = 1 | + 3-s + 5-s − 7-s + 15-s + 2·17-s + 19-s − 21-s − 27-s + 29-s − 35-s − 41-s + 2·51-s + 53-s + 57-s − 59-s + 2·71-s − 79-s − 81-s + 2·85-s + 87-s + 95-s − 105-s + 107-s − 2·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.009689496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.009689496\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - T + T^{2} \) |
| 5 | \( 1 - T + T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 - T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T + T^{2} \) |
| 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
−8.755410298271048498389308631901, −7.999417545792980813175618692824, −7.33917351977600529626997506197, −6.40267923867726936254234938719, −5.73990441558107138251824804161, −5.09686155646387229267659279621, −3.65558637434018811153982873261, −3.17519275614116072557691764724, −2.43484388670115121450925133596, −1.28075432621207545029060316135,
1.28075432621207545029060316135, 2.43484388670115121450925133596, 3.17519275614116072557691764724, 3.65558637434018811153982873261, 5.09686155646387229267659279621, 5.73990441558107138251824804161, 6.40267923867726936254234938719, 7.33917351977600529626997506197, 7.999417545792980813175618692824, 8.755410298271048498389308631901
