| L(s) = 1 | + 3-s − 0.776·5-s − 2.69·7-s + 9-s + 11-s + 13-s − 0.776·15-s + 2.62·17-s − 6.69·19-s − 2.69·21-s + 8.02·23-s − 4.39·25-s + 27-s − 8.09·29-s + 6.34·31-s + 33-s + 2.08·35-s − 0.398·37-s + 39-s + 6.87·41-s − 7.23·43-s − 0.776·45-s + 4.53·47-s + 0.245·49-s + 2.62·51-s − 2.75·53-s − 0.776·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.347·5-s − 1.01·7-s + 0.333·9-s + 0.301·11-s + 0.277·13-s − 0.200·15-s + 0.635·17-s − 1.53·19-s − 0.587·21-s + 1.67·23-s − 0.879·25-s + 0.192·27-s − 1.50·29-s + 1.13·31-s + 0.174·33-s + 0.353·35-s − 0.0655·37-s + 0.160·39-s + 1.07·41-s − 1.10·43-s − 0.115·45-s + 0.661·47-s + 0.0350·49-s + 0.367·51-s − 0.378·53-s − 0.104·55-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 | \( 1 + 0.776T + 5T^{2} \) |
| 7 | \( 1 + 2.69T + 7T^{2} \) |
| 17 | \( 1 - 2.62T + 17T^{2} \) |
| 19 | \( 1 + 6.69T + 19T^{2} \) |
| 23 | \( 1 - 8.02T + 23T^{2} \) |
| 29 | \( 1 + 8.09T + 29T^{2} \) |
| 31 | \( 1 - 6.34T + 31T^{2} \) |
| 37 | \( 1 + 0.398T + 37T^{2} \) |
| 41 | \( 1 - 6.87T + 41T^{2} \) |
| 43 | \( 1 + 7.23T + 43T^{2} \) |
| 47 | \( 1 - 4.53T + 47T^{2} \) |
| 53 | \( 1 + 2.75T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 5.02T + 61T^{2} \) |
| 67 | \( 1 - 4.95T + 67T^{2} \) |
| 71 | \( 1 - 5.96T + 71T^{2} \) |
| 73 | \( 1 - 3.79T + 73T^{2} \) |
| 79 | \( 1 + 2.14T + 79T^{2} \) |
| 83 | \( 1 + 3.73T + 83T^{2} \) |
| 89 | \( 1 - 2.37T + 89T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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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
−7.69931549120613894658678317727, −6.87043948374677563141267863912, −6.37335433894058034493883865681, −5.58514366414757080122352898665, −4.55672738539071443727940978206, −3.83929436225159118674327730523, −3.23427138365908541094591424137, −2.43824346715164929885908782318, −1.32224825706532631857483839484, 0,
1.32224825706532631857483839484, 2.43824346715164929885908782318, 3.23427138365908541094591424137, 3.83929436225159118674327730523, 4.55672738539071443727940978206, 5.58514366414757080122352898665, 6.37335433894058034493883865681, 6.87043948374677563141267863912, 7.69931549120613894658678317727
