| L(s) = 1 | + 3-s − 2.16·5-s + 3.64·7-s + 9-s + 11-s − 13-s − 2.16·15-s + 5.80·17-s − 2.70·19-s + 3.64·21-s − 9.50·23-s − 0.311·25-s + 27-s − 10.4·29-s − 10.5·31-s + 33-s − 7.88·35-s + 8.19·37-s − 39-s − 5.31·41-s + 8.74·43-s − 2.16·45-s − 5.41·47-s + 6.26·49-s + 5.80·51-s − 2·53-s − 2.16·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.968·5-s + 1.37·7-s + 0.333·9-s + 0.301·11-s − 0.277·13-s − 0.559·15-s + 1.40·17-s − 0.621·19-s + 0.794·21-s − 1.98·23-s − 0.0622·25-s + 0.192·27-s − 1.93·29-s − 1.88·31-s + 0.174·33-s − 1.33·35-s + 1.34·37-s − 0.160·39-s − 0.829·41-s + 1.33·43-s − 0.322·45-s − 0.790·47-s + 0.894·49-s + 0.813·51-s − 0.274·53-s − 0.291·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2.16T + 5T^{2} \) |
| 7 | \( 1 - 3.64T + 7T^{2} \) |
| 17 | \( 1 - 5.80T + 17T^{2} \) |
| 19 | \( 1 + 2.70T + 19T^{2} \) |
| 23 | \( 1 + 9.50T + 23T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 31 | \( 1 + 10.5T + 31T^{2} \) |
| 37 | \( 1 - 8.19T + 37T^{2} \) |
| 41 | \( 1 + 5.31T + 41T^{2} \) |
| 43 | \( 1 - 8.74T + 43T^{2} \) |
| 47 | \( 1 + 5.41T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 6.35T + 59T^{2} \) |
| 61 | \( 1 + 8.79T + 61T^{2} \) |
| 67 | \( 1 + 1.70T + 67T^{2} \) |
| 71 | \( 1 + 2.91T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 - 3.80T + 79T^{2} \) |
| 83 | \( 1 - 16.7T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 - 8.19T + 97T^{2} \) |
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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
−7.68165394261111518280434301041, −7.45373381675707415123634975112, −6.10440319603913174171597535515, −5.48650876704672013169035408176, −4.52038140080170433035025325220, −3.96743154727291190908728383430, −3.38500094392845237217112100946, −2.09834180615357068734978167867, −1.52950115257319226873209170406, 0,
1.52950115257319226873209170406, 2.09834180615357068734978167867, 3.38500094392845237217112100946, 3.96743154727291190908728383430, 4.52038140080170433035025325220, 5.48650876704672013169035408176, 6.10440319603913174171597535515, 7.45373381675707415123634975112, 7.68165394261111518280434301041
