| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s − 12-s − 15-s + 16-s − 18-s + 20-s + 2·23-s + 24-s + 25-s − 27-s + 30-s − 31-s − 32-s + 36-s − 40-s + 45-s − 2·46-s − 48-s − 49-s − 50-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s − 12-s − 15-s + 16-s − 18-s + 20-s + 2·23-s + 24-s + 25-s − 27-s + 30-s − 31-s − 32-s + 36-s − 40-s + 45-s − 2·46-s − 48-s − 49-s − 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6751355645\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6751355645\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( ( 1 - T )^{2} \) |
| 83 | \( ( 1 - T )^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 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
−9.413686380087340250337898507276, −8.961878643898988009518093885538, −7.79169520185332059722533843424, −6.93257982225243662865688566074, −6.43635886546139742682698002182, −5.55216903333981436184235212529, −4.89682525003054361210090925402, −3.32735351365026393663186551359, −2.09353453425380717254014201396, −1.06174209665756361827956558874,
1.06174209665756361827956558874, 2.09353453425380717254014201396, 3.32735351365026393663186551359, 4.89682525003054361210090925402, 5.55216903333981436184235212529, 6.43635886546139742682698002182, 6.93257982225243662865688566074, 7.79169520185332059722533843424, 8.961878643898988009518093885538, 9.413686380087340250337898507276
