| L(s) = 1 | + 2-s + 4-s + 8-s + 4·11-s + 2·13-s + 16-s + 17-s + 4·19-s + 4·22-s + 2·26-s + 10·29-s + 8·31-s + 32-s + 34-s + 2·37-s + 4·38-s − 10·41-s − 12·43-s + 4·44-s − 7·49-s + 2·52-s + 6·53-s + 10·58-s − 12·59-s − 10·61-s + 8·62-s + 64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1.20·11-s + 0.554·13-s + 1/4·16-s + 0.242·17-s + 0.917·19-s + 0.852·22-s + 0.392·26-s + 1.85·29-s + 1.43·31-s + 0.176·32-s + 0.171·34-s + 0.328·37-s + 0.648·38-s − 1.56·41-s − 1.82·43-s + 0.603·44-s − 49-s + 0.277·52-s + 0.824·53-s + 1.31·58-s − 1.56·59-s − 1.28·61-s + 1.01·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.104254605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.104254605\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.896056040959580640252932110069, −6.83414041172455162146867063165, −6.53203891348814136738759410716, −5.84474317529220730433927706238, −4.87609627522346518346123825914, −4.45515991381926668562779210131, −3.40072976272386003783075809662, −3.05379636583828922505383708373, −1.74861019337840755375028316172, −0.987259700426829860042560320630,
0.987259700426829860042560320630, 1.74861019337840755375028316172, 3.05379636583828922505383708373, 3.40072976272386003783075809662, 4.45515991381926668562779210131, 4.87609627522346518346123825914, 5.84474317529220730433927706238, 6.53203891348814136738759410716, 6.83414041172455162146867063165, 7.896056040959580640252932110069
