| L(s) = 1 | + 1.41·3-s + 14.1·5-s − 25·9-s − 54·11-s + 22.6·13-s + 20.0·15-s + 32.5·17-s + 63.6·19-s − 28·23-s + 75.0·25-s − 73.5·27-s + 282·29-s + 274.·31-s − 76.3·33-s + 146·37-s + 32.0·39-s + 340.·41-s − 10·43-s − 353.·45-s + 506.·47-s + 46·51-s + 598·53-s − 763.·55-s + 90.0·57-s − 575.·59-s + 466.·61-s + 320.·65-s + ⋯ |
| L(s) = 1 | + 0.272·3-s + 1.26·5-s − 0.925·9-s − 1.48·11-s + 0.482·13-s + 0.344·15-s + 0.464·17-s + 0.768·19-s − 0.253·23-s + 0.600·25-s − 0.524·27-s + 1.80·29-s + 1.58·31-s − 0.402·33-s + 0.648·37-s + 0.131·39-s + 1.29·41-s − 0.0354·43-s − 1.17·45-s + 1.57·47-s + 0.126·51-s + 1.54·53-s − 1.87·55-s + 0.209·57-s − 1.27·59-s + 0.979·61-s + 0.610·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.641269943\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.641269943\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 1.41T + 27T^{2} \) |
| 5 | \( 1 - 14.1T + 125T^{2} \) |
| 11 | \( 1 + 54T + 1.33e3T^{2} \) |
| 13 | \( 1 - 22.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 32.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 63.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 28T + 1.21e4T^{2} \) |
| 29 | \( 1 - 282T + 2.43e4T^{2} \) |
| 31 | \( 1 - 274.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 146T + 5.06e4T^{2} \) |
| 41 | \( 1 - 340.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 10T + 7.95e4T^{2} \) |
| 47 | \( 1 - 506.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 598T + 1.48e5T^{2} \) |
| 59 | \( 1 + 575.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 466.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 916T + 3.00e5T^{2} \) |
| 71 | \( 1 + 420T + 3.57e5T^{2} \) |
| 73 | \( 1 + 702.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 292T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.15e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 445.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 589.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.07948136604077186648816352785, −9.026088438692356610767422872042, −8.288751830811789895139330747830, −7.43953017072780290203024234002, −6.03089730716937934600077827307, −5.71599189048228048641878480063, −4.62523640626936360768743356442, −2.96111911798232945957984658801, −2.44964466447134845521646602494, −0.912410090285300200096663294185,
0.912410090285300200096663294185, 2.44964466447134845521646602494, 2.96111911798232945957984658801, 4.62523640626936360768743356442, 5.71599189048228048641878480063, 6.03089730716937934600077827307, 7.43953017072780290203024234002, 8.288751830811789895139330747830, 9.026088438692356610767422872042, 10.07948136604077186648816352785
