| L(s) = 1 | − 3-s − 2·7-s + 9-s + 4·11-s − 13-s − 6·17-s + 6·19-s + 2·21-s − 5·25-s − 27-s + 2·29-s + 6·31-s − 4·33-s − 10·37-s + 39-s + 8·41-s + 12·43-s − 12·47-s − 3·49-s + 6·51-s + 6·53-s − 6·57-s − 2·61-s − 2·63-s + 2·67-s + 8·71-s + 14·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 1.45·17-s + 1.37·19-s + 0.436·21-s − 25-s − 0.192·27-s + 0.371·29-s + 1.07·31-s − 0.696·33-s − 1.64·37-s + 0.160·39-s + 1.24·41-s + 1.82·43-s − 1.75·47-s − 3/7·49-s + 0.840·51-s + 0.824·53-s − 0.794·57-s − 0.256·61-s − 0.251·63-s + 0.244·67-s + 0.949·71-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.268600395\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.268600395\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−9.220283570203917667028022793155, −8.137716429689322885874117903176, −7.16333848096912753102386607562, −6.57091938769509480942354725697, −5.98247083912518480972422452649, −4.97254580700680097900229302596, −4.13958008081477749813044275306, −3.29289095796707662296197413885, −2.06399639050261723146787386632, −0.73510269161810698907992625597,
0.73510269161810698907992625597, 2.06399639050261723146787386632, 3.29289095796707662296197413885, 4.13958008081477749813044275306, 4.97254580700680097900229302596, 5.98247083912518480972422452649, 6.57091938769509480942354725697, 7.16333848096912753102386607562, 8.137716429689322885874117903176, 9.220283570203917667028022793155
