| L(s) = 1 | + 3-s − 5-s − 2·9-s − 3·11-s − 6·13-s − 15-s − 5·17-s − 19-s + 7·23-s − 4·25-s − 5·27-s + 2·29-s + 5·31-s − 3·33-s + 3·37-s − 6·39-s − 2·41-s + 4·43-s + 2·45-s − 5·47-s − 5·51-s − 53-s + 3·55-s − 57-s − 15·59-s − 5·61-s + 6·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 2/3·9-s − 0.904·11-s − 1.66·13-s − 0.258·15-s − 1.21·17-s − 0.229·19-s + 1.45·23-s − 4/5·25-s − 0.962·27-s + 0.371·29-s + 0.898·31-s − 0.522·33-s + 0.493·37-s − 0.960·39-s − 0.312·41-s + 0.609·43-s + 0.298·45-s − 0.729·47-s − 0.700·51-s − 0.137·53-s + 0.404·55-s − 0.132·57-s − 1.95·59-s − 0.640·61-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.740577269636355704704871435521, −8.997903461511478222170745270599, −8.095339691644100049496610049301, −7.49818629795120569559531103406, −6.48260699677724901373670959976, −5.20367487177393483929069503571, −4.45092229897982442279316013195, −3.02226969343536026770039863173, −2.34202066599205774134885333667, 0,
2.34202066599205774134885333667, 3.02226969343536026770039863173, 4.45092229897982442279316013195, 5.20367487177393483929069503571, 6.48260699677724901373670959976, 7.49818629795120569559531103406, 8.095339691644100049496610049301, 8.997903461511478222170745270599, 9.740577269636355704704871435521
