L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s − 2·10-s + 11-s + 2·13-s + 16-s − 6·17-s + 4·19-s − 2·20-s + 22-s + 6·23-s − 25-s + 2·26-s + 6·29-s − 2·31-s + 32-s − 6·34-s + 2·37-s + 4·38-s − 2·40-s − 10·41-s − 6·43-s + 44-s + 6·46-s − 50-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 0.632·10-s + 0.301·11-s + 0.554·13-s + 1/4·16-s − 1.45·17-s + 0.917·19-s − 0.447·20-s + 0.213·22-s + 1.25·23-s − 1/5·25-s + 0.392·26-s + 1.11·29-s − 0.359·31-s + 0.176·32-s − 1.02·34-s + 0.328·37-s + 0.648·38-s − 0.316·40-s − 1.56·41-s − 0.914·43-s + 0.150·44-s + 0.884·46-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.770938535\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.770938535\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−7.54824232068911395656064191370, −6.74805492397303772246489990919, −6.58364041888834181199164421811, −5.42804619739000563492874352692, −4.89526761664556889159659664564, −4.13729580279245842571558267901, −3.54695301666708082056930384176, −2.85146475476105833219768694809, −1.82174119142586825272219543863, −0.71419354043948938817251142086,
0.71419354043948938817251142086, 1.82174119142586825272219543863, 2.85146475476105833219768694809, 3.54695301666708082056930384176, 4.13729580279245842571558267901, 4.89526761664556889159659664564, 5.42804619739000563492874352692, 6.58364041888834181199164421811, 6.74805492397303772246489990919, 7.54824232068911395656064191370