| L(s) = 1 | + 3-s − 2·5-s + 9-s − 4·11-s + 13-s − 2·15-s − 6·17-s + 8·19-s − 8·23-s − 25-s + 27-s + 2·29-s − 8·31-s − 4·33-s − 10·37-s + 39-s + 6·41-s + 4·43-s − 2·45-s − 7·49-s − 6·51-s − 14·53-s + 8·55-s + 8·57-s + 12·59-s − 10·61-s − 2·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 0.516·15-s − 1.45·17-s + 1.83·19-s − 1.66·23-s − 1/5·25-s + 0.192·27-s + 0.371·29-s − 1.43·31-s − 0.696·33-s − 1.64·37-s + 0.160·39-s + 0.937·41-s + 0.609·43-s − 0.298·45-s − 49-s − 0.840·51-s − 1.92·53-s + 1.07·55-s + 1.05·57-s + 1.56·59-s − 1.28·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.231246651526005478557570132192, −8.342770632372487309297815864648, −7.72597071702691609645308173983, −7.13855453193683225508653499024, −5.90056466305320169868428843649, −4.90251153820633880270104247107, −3.93592460575099925248016367060, −3.10657645810316901845317440603, −1.93631023376644655386209086893, 0,
1.93631023376644655386209086893, 3.10657645810316901845317440603, 3.93592460575099925248016367060, 4.90251153820633880270104247107, 5.90056466305320169868428843649, 7.13855453193683225508653499024, 7.72597071702691609645308173983, 8.342770632372487309297815864648, 9.231246651526005478557570132192
