L(s) = 1 | + 5-s + 2·7-s + 2·11-s + 2·13-s + 8·19-s + 8·23-s + 25-s + 2·29-s + 2·35-s − 8·37-s − 8·41-s + 43-s − 3·49-s + 2·53-s + 2·55-s − 6·59-s + 8·61-s + 2·65-s − 12·67-s − 2·73-s + 4·77-s + 12·79-s + 18·83-s + 6·89-s + 4·91-s + 8·95-s − 2·97-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.755·7-s + 0.603·11-s + 0.554·13-s + 1.83·19-s + 1.66·23-s + 1/5·25-s + 0.371·29-s + 0.338·35-s − 1.31·37-s − 1.24·41-s + 0.152·43-s − 3/7·49-s + 0.274·53-s + 0.269·55-s − 0.781·59-s + 1.02·61-s + 0.248·65-s − 1.46·67-s − 0.234·73-s + 0.455·77-s + 1.35·79-s + 1.97·83-s + 0.635·89-s + 0.419·91-s + 0.820·95-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.854757002\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.854757002\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.17088888222688, −14.41075581438731, −14.12323928519976, −13.48684821972916, −13.21696771825167, −12.33635103068448, −11.80677854293196, −11.49541357774791, −10.76197269745860, −10.39552558286156, −9.609415446927807, −9.122454126967912, −8.699540585980383, −8.015931260834763, −7.369736344861774, −6.853724904613722, −6.270127752529577, −5.442522935836509, −5.069193591566051, −4.514354246513125, −3.361417678653131, −3.272688847403610, −2.098330332219115, −1.398157917038380, −0.8361772640463910,
0.8361772640463910, 1.398157917038380, 2.098330332219115, 3.272688847403610, 3.361417678653131, 4.514354246513125, 5.069193591566051, 5.442522935836509, 6.270127752529577, 6.853724904613722, 7.369736344861774, 8.015931260834763, 8.699540585980383, 9.122454126967912, 9.609415446927807, 10.39552558286156, 10.76197269745860, 11.49541357774791, 11.80677854293196, 12.33635103068448, 13.21696771825167, 13.48684821972916, 14.12323928519976, 14.41075581438731, 15.17088888222688