| L(s) = 1 | + 5-s + 4·7-s + 4·11-s + 6·13-s − 2·17-s − 4·19-s + 25-s − 10·29-s + 4·31-s + 4·35-s − 10·37-s − 2·41-s + 4·43-s + 8·47-s + 9·49-s − 2·53-s + 4·55-s + 12·59-s − 10·61-s + 6·65-s − 12·67-s + 10·73-s + 16·77-s + 4·79-s + 4·83-s − 2·85-s + 6·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s + 1.20·11-s + 1.66·13-s − 0.485·17-s − 0.917·19-s + 1/5·25-s − 1.85·29-s + 0.718·31-s + 0.676·35-s − 1.64·37-s − 0.312·41-s + 0.609·43-s + 1.16·47-s + 9/7·49-s − 0.274·53-s + 0.539·55-s + 1.56·59-s − 1.28·61-s + 0.744·65-s − 1.46·67-s + 1.17·73-s + 1.82·77-s + 0.450·79-s + 0.439·83-s − 0.216·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.417890213\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.417890213\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 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
−9.258401000083574088505848754263, −8.760125119461013693220157342202, −8.116410983938513625426203426805, −7.03872324148136889156556818396, −6.21305657818228885677430783028, −5.44404884721337967389713010913, −4.36333484257300403943968845908, −3.69324260965892944900735137760, −2.02785893596435789244461953313, −1.31634970517745226225569387554,
1.31634970517745226225569387554, 2.02785893596435789244461953313, 3.69324260965892944900735137760, 4.36333484257300403943968845908, 5.44404884721337967389713010913, 6.21305657818228885677430783028, 7.03872324148136889156556818396, 8.116410983938513625426203426805, 8.760125119461013693220157342202, 9.258401000083574088505848754263
