| L(s) = 1 | + 3·3-s − 6·5-s + 7·7-s + 9·9-s − 36·11-s − 62·13-s − 18·15-s + 114·17-s + 76·19-s + 21·21-s − 24·23-s − 89·25-s + 27·27-s − 54·29-s − 112·31-s − 108·33-s − 42·35-s + 178·37-s − 186·39-s + 378·41-s + 172·43-s − 54·45-s − 192·47-s + 49·49-s + 342·51-s + 402·53-s + 216·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.536·5-s + 0.377·7-s + 1/3·9-s − 0.986·11-s − 1.32·13-s − 0.309·15-s + 1.62·17-s + 0.917·19-s + 0.218·21-s − 0.217·23-s − 0.711·25-s + 0.192·27-s − 0.345·29-s − 0.648·31-s − 0.569·33-s − 0.202·35-s + 0.790·37-s − 0.763·39-s + 1.43·41-s + 0.609·43-s − 0.178·45-s − 0.595·47-s + 1/7·49-s + 0.939·51-s + 1.04·53-s + 0.529·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.160167978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.160167978\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| good | 5 | \( 1 + 6 T + p^{3} T^{2} \) |
| 11 | \( 1 + 36 T + p^{3} T^{2} \) |
| 13 | \( 1 + 62 T + p^{3} T^{2} \) |
| 17 | \( 1 - 114 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 23 | \( 1 + 24 T + p^{3} T^{2} \) |
| 29 | \( 1 + 54 T + p^{3} T^{2} \) |
| 31 | \( 1 + 112 T + p^{3} T^{2} \) |
| 37 | \( 1 - 178 T + p^{3} T^{2} \) |
| 41 | \( 1 - 378 T + p^{3} T^{2} \) |
| 43 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 47 | \( 1 + 192 T + p^{3} T^{2} \) |
| 53 | \( 1 - 402 T + p^{3} T^{2} \) |
| 59 | \( 1 + 396 T + p^{3} T^{2} \) |
| 61 | \( 1 + 254 T + p^{3} T^{2} \) |
| 67 | \( 1 - 1012 T + p^{3} T^{2} \) |
| 71 | \( 1 - 840 T + p^{3} T^{2} \) |
| 73 | \( 1 - 890 T + p^{3} T^{2} \) |
| 79 | \( 1 - 80 T + p^{3} T^{2} \) |
| 83 | \( 1 - 108 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1638 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1010 T + p^{3} T^{2} \) |
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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
−9.459161986228085718360889151234, −8.066744689152111430519350880002, −7.77655579038787733494575565297, −7.20467830621980351953848978888, −5.70132038002626292919469950072, −5.06161112421647104886201793207, −3.99647932369199246792330324316, −3.05012282855403358399382246415, −2.14133594036598931656883575836, −0.70446634354166777485363125529,
0.70446634354166777485363125529, 2.14133594036598931656883575836, 3.05012282855403358399382246415, 3.99647932369199246792330324316, 5.06161112421647104886201793207, 5.70132038002626292919469950072, 7.20467830621980351953848978888, 7.77655579038787733494575565297, 8.066744689152111430519350880002, 9.459161986228085718360889151234
