| L(s) = 1 | + 3-s + 3·7-s − 2·9-s − 2·11-s − 13-s + 5·17-s − 19-s + 3·21-s − 23-s − 5·27-s − 3·29-s − 4·31-s − 2·33-s − 2·37-s − 39-s − 8·41-s − 8·43-s − 8·47-s + 2·49-s + 5·51-s − 9·53-s − 57-s − 59-s + 14·61-s − 6·63-s + 13·67-s − 69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.13·7-s − 2/3·9-s − 0.603·11-s − 0.277·13-s + 1.21·17-s − 0.229·19-s + 0.654·21-s − 0.208·23-s − 0.962·27-s − 0.557·29-s − 0.718·31-s − 0.348·33-s − 0.328·37-s − 0.160·39-s − 1.24·41-s − 1.21·43-s − 1.16·47-s + 2/7·49-s + 0.700·51-s − 1.23·53-s − 0.132·57-s − 0.130·59-s + 1.79·61-s − 0.755·63-s + 1.58·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.72570962465753433033134171054, −7.03746081834115821689444638722, −6.03489966823382060239173324440, −5.22893948551161166678750535882, −4.94047949903662554661829583018, −3.73594259498935797345355690698, −3.16800606098036868848903599918, −2.19695790616163302044018829314, −1.51234840449082039342462376961, 0,
1.51234840449082039342462376961, 2.19695790616163302044018829314, 3.16800606098036868848903599918, 3.73594259498935797345355690698, 4.94047949903662554661829583018, 5.22893948551161166678750535882, 6.03489966823382060239173324440, 7.03746081834115821689444638722, 7.72570962465753433033134171054
