| L(s) = 1 | − 5·3-s − 7·7-s − 2·9-s + 11·11-s + 46·13-s − 127·17-s + 117·19-s + 35·21-s − 80·23-s + 145·27-s + 34·29-s − 292·31-s − 55·33-s + 376·37-s − 230·39-s + 507·41-s − 32·43-s + 134·47-s + 49·49-s + 635·51-s − 612·53-s − 585·57-s + 780·59-s − 426·61-s + 14·63-s + 207·67-s + 400·69-s + ⋯ |
| L(s) = 1 | − 0.962·3-s − 0.377·7-s − 0.0740·9-s + 0.301·11-s + 0.981·13-s − 1.81·17-s + 1.41·19-s + 0.363·21-s − 0.725·23-s + 1.03·27-s + 0.217·29-s − 1.69·31-s − 0.290·33-s + 1.67·37-s − 0.944·39-s + 1.93·41-s − 0.113·43-s + 0.415·47-s + 1/7·49-s + 1.74·51-s − 1.58·53-s − 1.35·57-s + 1.72·59-s − 0.894·61-s + 0.0279·63-s + 0.377·67-s + 0.697·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 3 | \( 1 + 5 T + p^{3} T^{2} \) |
| 11 | \( 1 - p T + p^{3} T^{2} \) |
| 13 | \( 1 - 46 T + p^{3} T^{2} \) |
| 17 | \( 1 + 127 T + p^{3} T^{2} \) |
| 19 | \( 1 - 117 T + p^{3} T^{2} \) |
| 23 | \( 1 + 80 T + p^{3} T^{2} \) |
| 29 | \( 1 - 34 T + p^{3} T^{2} \) |
| 31 | \( 1 + 292 T + p^{3} T^{2} \) |
| 37 | \( 1 - 376 T + p^{3} T^{2} \) |
| 41 | \( 1 - 507 T + p^{3} T^{2} \) |
| 43 | \( 1 + 32 T + p^{3} T^{2} \) |
| 47 | \( 1 - 134 T + p^{3} T^{2} \) |
| 53 | \( 1 + 612 T + p^{3} T^{2} \) |
| 59 | \( 1 - 780 T + p^{3} T^{2} \) |
| 61 | \( 1 + 426 T + p^{3} T^{2} \) |
| 67 | \( 1 - 207 T + p^{3} T^{2} \) |
| 71 | \( 1 - 702 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1185 T + p^{3} T^{2} \) |
| 79 | \( 1 - 54 T + p^{3} T^{2} \) |
| 83 | \( 1 - 309 T + p^{3} T^{2} \) |
| 89 | \( 1 + 339 T + p^{3} T^{2} \) |
| 97 | \( 1 + 182 T + p^{3} T^{2} \) |
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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
−8.984218746631168725093820501434, −7.937023125521643796181879560087, −6.96360799110053425974050726074, −6.16649262451683966107240814147, −5.68670401290049113344842795894, −4.59432666505972550291966343128, −3.71170341072324909589946783121, −2.49936899088387744022664138198, −1.10025342141716840847961775483, 0,
1.10025342141716840847961775483, 2.49936899088387744022664138198, 3.71170341072324909589946783121, 4.59432666505972550291966343128, 5.68670401290049113344842795894, 6.16649262451683966107240814147, 6.96360799110053425974050726074, 7.937023125521643796181879560087, 8.984218746631168725093820501434
