| L(s) = 1 | + 5-s + 4·7-s − 11-s + 13-s − 2·19-s + 6·23-s + 25-s − 6·29-s − 2·31-s + 4·35-s − 4·37-s + 6·41-s − 8·43-s + 9·49-s − 55-s − 12·59-s + 2·61-s + 65-s − 8·67-s + 12·71-s + 8·73-s − 4·77-s − 8·79-s + 12·83-s − 6·89-s + 4·91-s − 2·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s − 0.301·11-s + 0.277·13-s − 0.458·19-s + 1.25·23-s + 1/5·25-s − 1.11·29-s − 0.359·31-s + 0.676·35-s − 0.657·37-s + 0.937·41-s − 1.21·43-s + 9/7·49-s − 0.134·55-s − 1.56·59-s + 0.256·61-s + 0.124·65-s − 0.977·67-s + 1.42·71-s + 0.936·73-s − 0.455·77-s − 0.900·79-s + 1.31·83-s − 0.635·89-s + 0.419·91-s − 0.205·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.92855817904547, −13.59572323171035, −12.94065186431549, −12.63673213516566, −11.95938972802089, −11.40271395500409, −10.96395742189601, −10.75020984884705, −10.12544851408019, −9.416532281198550, −8.979362120961277, −8.575981239141368, −7.831578863063001, −7.699013677097880, −6.894800410625413, −6.456096615929593, −5.669711037583803, −5.246888199808226, −4.866099943211486, −4.227328232676651, −3.592279534641631, −2.856215498423786, −2.166917966962091, −1.619432893594030, −1.092859582368949, 0,
1.092859582368949, 1.619432893594030, 2.166917966962091, 2.856215498423786, 3.592279534641631, 4.227328232676651, 4.866099943211486, 5.246888199808226, 5.669711037583803, 6.456096615929593, 6.894800410625413, 7.699013677097880, 7.831578863063001, 8.575981239141368, 8.979362120961277, 9.416532281198550, 10.12544851408019, 10.75020984884705, 10.96395742189601, 11.40271395500409, 11.95938972802089, 12.63673213516566, 12.94065186431549, 13.59572323171035, 13.92855817904547
