L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 2·11-s + 6·13-s + 14-s + 16-s − 5·17-s + 6·19-s − 2·22-s + 3·23-s − 6·26-s − 28-s + 6·29-s − 4·31-s − 32-s + 5·34-s − 4·37-s − 6·38-s + 41-s + 7·43-s + 2·44-s − 3·46-s − 2·47-s + 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 − 1.21·17-s + 1.37·19-s − 0.426·22-s + 0.625·23-s − 1.17·26-s − 0.188·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s + 0.857·34-s − 0.657·37-s − 0.973·38-s + 0.156·41-s + 1.06·43-s + 0.301·44-s − 0.442·46-s − 0.291·47-s + 1/7·49-s + 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.678187248\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.678187248\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 17 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 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
−7.73128340478956900314019234734, −6.95831101671641335743294068905, −6.51342434993942828121705922760, −5.85355769410195425348113483701, −5.03367447047149124934006214406, −3.99558406086110068391045283643, −3.42067138801555542578717234379, −2.54919024232704436677704452597, −1.47966520692094645798048175755, −0.74967372757513971521162401395,
0.74967372757513971521162401395, 1.47966520692094645798048175755, 2.54919024232704436677704452597, 3.42067138801555542578717234379, 3.99558406086110068391045283643, 5.03367447047149124934006214406, 5.85355769410195425348113483701, 6.51342434993942828121705922760, 6.95831101671641335743294068905, 7.73128340478956900314019234734