| L(s) = 1 | + 5-s + 7-s + 9-s − 17-s − 23-s + 25-s − 29-s + 31-s + 35-s − 37-s − 41-s − 2·43-s + 45-s − 53-s + 59-s + 63-s + 67-s + 71-s + 81-s + 83-s − 85-s + 2·97-s − 101-s − 2·103-s + 107-s − 113-s − 115-s + ⋯ |
| L(s) = 1 | + 5-s + 7-s + 9-s − 17-s − 23-s + 25-s − 29-s + 31-s + 35-s − 37-s − 41-s − 2·43-s + 45-s − 53-s + 59-s + 63-s + 67-s + 71-s + 81-s + 83-s − 85-s + 2·97-s − 101-s − 2·103-s + 107-s − 113-s − 115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.549350541\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.549350541\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 + T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )^{2} \) |
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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
−9.581105540897436401233257443107, −8.610218294283092694995096963078, −7.988529284824344838276495522214, −6.92077431313453624065434922360, −6.38675167851080530288631223688, −5.23075468928305258882374356700, −4.71669038447938970781885792438, −3.65255972961712658432044019576, −2.16036689923691567904060690391, −1.57610827122058409791962956815,
1.57610827122058409791962956815, 2.16036689923691567904060690391, 3.65255972961712658432044019576, 4.71669038447938970781885792438, 5.23075468928305258882374356700, 6.38675167851080530288631223688, 6.92077431313453624065434922360, 7.988529284824344838276495522214, 8.610218294283092694995096963078, 9.581105540897436401233257443107
