L(s) = 1 | − 5-s + 2·7-s + 2·13-s − 2·19-s + 25-s − 8·31-s − 2·35-s − 2·37-s − 2·43-s − 3·49-s + 6·53-s + 12·59-s + 2·61-s − 2·65-s − 4·67-s − 2·73-s − 10·79-s − 12·83-s + 6·89-s + 4·91-s + 2·95-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.755·7-s + 0.554·13-s − 0.458·19-s + 1/5·25-s − 1.43·31-s − 0.338·35-s − 0.328·37-s − 0.304·43-s − 3/7·49-s + 0.824·53-s + 1.56·59-s + 0.256·61-s − 0.248·65-s − 0.488·67-s − 0.234·73-s − 1.12·79-s − 1.31·83-s + 0.635·89-s + 0.419·91-s + 0.205·95-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.860832098\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.860832098\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−12.58070086066876, −11.97841391353438, −11.60014328175676, −11.25846187678970, −10.80456686295464, −10.35019917552988, −9.949978044591141, −9.187004837247783, −8.892087872187767, −8.390940070509748, −8.037273591973061, −7.513253923940526, −7.013244216913822, −6.642281765557586, −5.957405740511835, −5.428386504848949, −5.141954570141147, −4.309711040347916, −4.139614307457958, −3.476848389824674, −2.987299991794290, −2.205379828976531, −1.746168947412926, −1.140803446789325, −0.3703842743846824,
0.3703842743846824, 1.140803446789325, 1.746168947412926, 2.205379828976531, 2.987299991794290, 3.476848389824674, 4.139614307457958, 4.309711040347916, 5.141954570141147, 5.428386504848949, 5.957405740511835, 6.642281765557586, 7.013244216913822, 7.513253923940526, 8.037273591973061, 8.390940070509748, 8.892087872187767, 9.187004837247783, 9.949978044591141, 10.35019917552988, 10.80456686295464, 11.25846187678970, 11.60014328175676, 11.97841391353438, 12.58070086066876