| L(s) = 1 | + 3-s + 5·7-s − 2·9-s + 5·11-s + 13-s − 4·17-s + 4·19-s + 5·21-s + 9·23-s − 5·25-s − 5·27-s + 2·29-s − 2·31-s + 5·33-s − 6·37-s + 39-s + 6·41-s + 5·43-s − 47-s + 18·49-s − 4·51-s − 6·53-s + 4·57-s + 12·59-s − 3·61-s − 10·63-s + 5·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.88·7-s − 2/3·9-s + 1.50·11-s + 0.277·13-s − 0.970·17-s + 0.917·19-s + 1.09·21-s + 1.87·23-s − 25-s − 0.962·27-s + 0.371·29-s − 0.359·31-s + 0.870·33-s − 0.986·37-s + 0.160·39-s + 0.937·41-s + 0.762·43-s − 0.145·47-s + 18/7·49-s − 0.560·51-s − 0.824·53-s + 0.529·57-s + 1.56·59-s − 0.384·61-s − 1.25·63-s + 0.610·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.696016402\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.696016402\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.87486373206769811831775045563, −7.29750014370213526577280459559, −6.54249061350008038886521602658, −5.61444554737392916357790744857, −5.01550680814523549332081775649, −4.24280458253081682699218969360, −3.58415977417114514479354141880, −2.57990639865260513859150699397, −1.74227584587096820250846727663, −1.01247445303881114965200489091,
1.01247445303881114965200489091, 1.74227584587096820250846727663, 2.57990639865260513859150699397, 3.58415977417114514479354141880, 4.24280458253081682699218969360, 5.01550680814523549332081775649, 5.61444554737392916357790744857, 6.54249061350008038886521602658, 7.29750014370213526577280459559, 7.87486373206769811831775045563
