| L(s) = 1 | + 158·3-s − 625·5-s + 1.92e3·7-s + 1.84e4·9-s − 9.87e4·15-s + 3.03e5·21-s + 2.11e5·23-s + 3.90e5·25-s + 1.87e6·27-s − 2.06e4·29-s − 1.20e6·35-s − 5.41e6·41-s + 2.51e6·43-s − 1.15e7·45-s + 9.61e6·47-s − 2.07e6·49-s + 1.10e7·61-s + 3.53e7·63-s + 2.02e7·67-s + 3.33e7·69-s + 6.17e7·75-s + 1.74e8·81-s + 3.08e7·83-s − 3.26e6·87-s − 1.06e8·89-s + 1.77e8·101-s + 1.46e8·103-s + ⋯ |
| L(s) = 1 | + 1.95·3-s − 5-s + 0.800·7-s + 2.80·9-s − 1.95·15-s + 1.56·21-s + 0.754·23-s + 25-s + 3.52·27-s − 0.0291·29-s − 0.800·35-s − 1.91·41-s + 0.736·43-s − 2.80·45-s + 1.97·47-s − 0.359·49-s + 0.798·61-s + 2.24·63-s + 1.00·67-s + 1.47·69-s + 1.95·75-s + 4.06·81-s + 0.650·83-s − 0.0569·87-s − 1.70·89-s + 1.70·101-s + 1.30·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(5.044348935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.044348935\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p^{4} T \) |
| good | 3 | \( 1 - 158 T + p^{8} T^{2} \) |
| 7 | \( 1 - 1922 T + p^{8} T^{2} \) |
| 11 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 13 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 17 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 19 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 23 | \( 1 - 211202 T + p^{8} T^{2} \) |
| 29 | \( 1 + 20642 T + p^{8} T^{2} \) |
| 31 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 37 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 41 | \( 1 + 5419198 T + p^{8} T^{2} \) |
| 43 | \( 1 - 2519518 T + p^{8} T^{2} \) |
| 47 | \( 1 - 9618242 T + p^{8} T^{2} \) |
| 53 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 59 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 61 | \( 1 - 11061598 T + p^{8} T^{2} \) |
| 67 | \( 1 - 20249758 T + p^{8} T^{2} \) |
| 71 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 73 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 79 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 83 | \( 1 - 30884638 T + p^{8} T^{2} \) |
| 89 | \( 1 + 106804798 T + p^{8} T^{2} \) |
| 97 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
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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
−10.06617710321044129787147971403, −8.937967979500463089915382907390, −8.416089485616362485690783155929, −7.63376275203541595423411397854, −6.96225800122544500029939440765, −4.88998281517618667767103937489, −3.99361436794890804529440289444, −3.16179533194050274124056986071, −2.11417562067079059331440748298, −0.984767586116670727188127507535,
0.984767586116670727188127507535, 2.11417562067079059331440748298, 3.16179533194050274124056986071, 3.99361436794890804529440289444, 4.88998281517618667767103937489, 6.96225800122544500029939440765, 7.63376275203541595423411397854, 8.416089485616362485690783155929, 8.937967979500463089915382907390, 10.06617710321044129787147971403
