| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 2·11-s − 6·13-s − 14-s + 16-s − 2·17-s + 6·19-s − 2·22-s + 23-s − 6·26-s − 28-s + 9·29-s − 2·31-s + 32-s − 2·34-s + 2·37-s + 6·38-s − 11·41-s − 4·43-s − 2·44-s + 46-s − 7·47-s − 6·49-s − 6·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 0.603·11-s − 1.66·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s + 1.37·19-s − 0.426·22-s + 0.208·23-s − 1.17·26-s − 0.188·28-s + 1.67·29-s − 0.359·31-s + 0.176·32-s − 0.342·34-s + 0.328·37-s + 0.973·38-s − 1.71·41-s − 0.609·43-s − 0.301·44-s + 0.147·46-s − 1.02·47-s − 6/7·49-s − 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.88514030252187489642033509711, −7.20434253837491676278095910777, −6.65833186650970181505726142694, −5.71020512407381805376716063928, −4.92635424763021168885568557735, −4.54242851420671311383585730557, −3.14601060022478743722281557456, −2.84280000179388060239463653961, −1.63049357083878420442545758226, 0,
1.63049357083878420442545758226, 2.84280000179388060239463653961, 3.14601060022478743722281557456, 4.54242851420671311383585730557, 4.92635424763021168885568557735, 5.71020512407381805376716063928, 6.65833186650970181505726142694, 7.20434253837491676278095910777, 7.88514030252187489642033509711
