| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s + 9-s − 10-s − 12-s + 13-s + 15-s + 16-s + 18-s − 20-s − 24-s + 25-s + 26-s − 27-s + 30-s + 32-s + 36-s − 39-s − 40-s + 2·41-s + 2·43-s − 45-s − 48-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s + 9-s − 10-s − 12-s + 13-s + 15-s + 16-s + 18-s − 20-s − 24-s + 25-s + 26-s − 27-s + 30-s + 32-s + 36-s − 39-s − 40-s + 2·41-s + 2·43-s − 45-s − 48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.421141925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.421141925\) |
| \(L(1)\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )^{2} \) |
| 43 | \( ( 1 - T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 + T )^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( ( 1 + T )^{2} \) |
| 83 | \( ( 1 + T )^{2} \) |
| 89 | \( ( 1 + T )^{2} \) |
| 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
−9.931554638273020996927713404470, −8.698255908362668068946069796125, −7.65246864494546579889506621795, −7.12037807260121944998443139258, −6.14698985073584382172958062002, −5.60367531171539076083974330184, −4.42612631086730911928806235988, −4.06854689343663696633089946560, −2.89886796438064589474271164547, −1.25790435159419995088458449673,
1.25790435159419995088458449673, 2.89886796438064589474271164547, 4.06854689343663696633089946560, 4.42612631086730911928806235988, 5.60367531171539076083974330184, 6.14698985073584382172958062002, 7.12037807260121944998443139258, 7.65246864494546579889506621795, 8.698255908362668068946069796125, 9.931554638273020996927713404470
