| L(s) = 1 | − 3-s − 3·5-s + 9-s − 11-s + 4·13-s + 3·15-s − 4·17-s + 8·23-s + 4·25-s − 27-s − 7·29-s − 11·31-s + 33-s + 4·37-s − 4·39-s + 4·41-s − 2·43-s − 3·45-s + 2·47-s + 4·51-s − 11·53-s + 3·55-s − 7·59-s − 10·61-s − 12·65-s + 10·67-s − 8·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 1/3·9-s − 0.301·11-s + 1.10·13-s + 0.774·15-s − 0.970·17-s + 1.66·23-s + 4/5·25-s − 0.192·27-s − 1.29·29-s − 1.97·31-s + 0.174·33-s + 0.657·37-s − 0.640·39-s + 0.624·41-s − 0.304·43-s − 0.447·45-s + 0.291·47-s + 0.560·51-s − 1.51·53-s + 0.404·55-s − 0.911·59-s − 1.28·61-s − 1.48·65-s + 1.22·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8177441996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8177441996\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 11 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−8.196884566769907012523530039680, −7.51155936293345874727212587536, −6.97146397688082967666818404984, −6.14168801835031951132380208285, −5.32244844888582985746142827540, −4.52864700037723516870251514543, −3.81859118303297002295385754293, −3.14314898901266199472959204646, −1.75936318150012083215616202954, −0.51542501758314193126777624873,
0.51542501758314193126777624873, 1.75936318150012083215616202954, 3.14314898901266199472959204646, 3.81859118303297002295385754293, 4.52864700037723516870251514543, 5.32244844888582985746142827540, 6.14168801835031951132380208285, 6.97146397688082967666818404984, 7.51155936293345874727212587536, 8.196884566769907012523530039680
