| L(s) = 1 | − 2.30·2-s + 3.30·4-s + 2.60·7-s − 3.00·8-s − 4.60·11-s − 6.60·13-s − 6·14-s + 0.302·16-s + 1.60·17-s + 3.60·19-s + 10.6·22-s − 3·23-s + 15.2·26-s + 8.60·28-s − 1.39·29-s − 5.60·31-s + 5.30·32-s − 3.69·34-s − 2·37-s − 8.30·38-s + 4.60·41-s − 0.605·43-s − 15.2·44-s + 6.90·46-s − 9.21·47-s − 0.211·49-s − 21.8·52-s + ⋯ |
| L(s) = 1 | − 1.62·2-s + 1.65·4-s + 0.984·7-s − 1.06·8-s − 1.38·11-s − 1.83·13-s − 1.60·14-s + 0.0756·16-s + 0.389·17-s + 0.827·19-s + 2.26·22-s − 0.625·23-s + 2.98·26-s + 1.62·28-s − 0.258·29-s − 1.00·31-s + 0.937·32-s − 0.634·34-s − 0.328·37-s − 1.34·38-s + 0.719·41-s − 0.0923·43-s − 2.29·44-s + 1.01·46-s − 1.34·47-s − 0.0301·49-s − 3.02·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 7 | \( 1 - 2.60T + 7T^{2} \) |
| 11 | \( 1 + 4.60T + 11T^{2} \) |
| 13 | \( 1 + 6.60T + 13T^{2} \) |
| 17 | \( 1 - 1.60T + 17T^{2} \) |
| 19 | \( 1 - 3.60T + 19T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 + 1.39T + 29T^{2} \) |
| 31 | \( 1 + 5.60T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 4.60T + 41T^{2} \) |
| 43 | \( 1 + 0.605T + 43T^{2} \) |
| 47 | \( 1 + 9.21T + 47T^{2} \) |
| 53 | \( 1 - 1.60T + 53T^{2} \) |
| 59 | \( 1 - 1.39T + 59T^{2} \) |
| 61 | \( 1 + 4.21T + 61T^{2} \) |
| 67 | \( 1 - 0.788T + 67T^{2} \) |
| 71 | \( 1 + 7.39T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 - 3T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−9.988107137886650927247968358669, −9.309797524028500478438069796566, −8.199124829389745687943754952952, −7.67108039533617462514262826181, −7.16056784530961885576662685931, −5.55035658178142213457820676917, −4.72360205450387309207626994421, −2.76553979804537263980696040981, −1.74952470977637072478622222235, 0,
1.74952470977637072478622222235, 2.76553979804537263980696040981, 4.72360205450387309207626994421, 5.55035658178142213457820676917, 7.16056784530961885576662685931, 7.67108039533617462514262826181, 8.199124829389745687943754952952, 9.309797524028500478438069796566, 9.988107137886650927247968358669
